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University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Masters Theses Graduate School

5-2012

A High-Energy Neutron Flux Spectra Measurement Method for the Spallation Neutron Source

Nicholas Patrick Luciano [email protected]

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Recommended Citation Luciano, Nicholas Patrick, "A High-Energy Neutron Flux Spectra Measurement Method for the Spallation Neutron Source. " Master's Thesis, University of Tennessee, 2012. https://trace.tennessee.edu/utk_gradthes/1178

This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a thesis written by Nicholas Patrick Luciano entitled "A High-Energy Neutron Flux Spectra Measurement Method for the Spallation Neutron Source." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Nuclear Engineering.

Lawrence F. Miller, Major Professor

We have read this thesis and recommend its acceptance:

Erik B. Iverson, Ronald E. Pevey

Accepted for the Council: Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) A High-Energy Neutron Flux Spectra Measurement Method for the Spallation Neutron Source

A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville

ii Dedication

What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from...

We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the ﬁrst time.

–T.S. Eliot, Little Gidding

iii Acknowledgments

This work would not have been possible without the support of many. I must thank Dr. Erik Iverson, my advisor and mentor, for dedicating innumerable hours to my education. Erik, I will consider myself quite accomplished if someday I possess even a fraction of your knowledge, experience, and skill. Also, I would like to thank my committee members Dr. Larry Miller and Dr. Ron Pevey, who both have taught me much. The staff members of the Spallation Neutron Source are too many to name, but I will try to list a few: the neutronics group - Phil Ferguson, Franz Gallmeier, Wei Lu, Ceris Hamilton, and Wendy Brooks; the beamline staff - Doug Abernathy, Matt Stone, Ashfia Huq, and Jason Hodges; David Freeman; Pedro Gonzalez, David Craft, Jim Treen, and the RCTs; the instrument support staff - especially David Dunning, Ray Wilson, and Jack Frye; George Dodson, Larry Longcoy, and the accelerator operations staff. To those that I’ve neglected to list here, you have my apologies and my sincere thanks. Mom, Dad, Jessi, and Dustin are the constants in my life that enable me to explore the variables. I can only offer you my love and thanks in return. Finally, to my fiancée Jennifer Niedziela, words cannot express how much I appreciate you. You are my best friend... and while doing this work, it was especially nice to have a best friend who knew a little something about neutrons. Thank you, I love you, and I’m looking forward to our life together.

iv Abstract

The goal of this work was to develop a foil activation method to measure high-energy (1–120 MeV) neutron flux spectra at the Spallation Neutron Source by researching the scientific literature, assembling an experimental apparatus, performing experiments, analyzing the results, and refining the technique based on experience. The primary motivation for this work is to provide a benchmark for the neutron source term used in target station and shielding simulations Two sets of foil irradiations were performed, one at the ARCS beamline and one at the POWGEN beamline. The gamma radiation of the foil activation products was measured with a high purity germanium gamma-ray spectrometer, and the product reaction rates during irradiation were quantified. Corrections, such as self-shielding factors, were applied to the measurements to account for particular effects. The corrected measurement data, along with calculated response functions and an initial guess spectrum, were input to the MAXED neutron spectrum unfolding computer code. MAXED uses the maximum entropy method to unfold an output spectrum that is the minimally modified guess spectrum consistent with the measurement data. The foil irradiation and subsequent analysis from the ARCS spectrum produced a reasonable neutron spectrum, which noticeably differed from the initial guess spectrum. This measurement is regarded as consistent, but yet unverified. The gamma-ray spectrum of the foil irradiation at the POWGEN beamline showed no high-energy activation. This is regarded as an experimental error, and no conclusions can be drawn about the high-energy neutron spectrum. Future foil irradiations are planned to verify and expand the neutron spectrum measurements.

v Contents

1 Introduction1 1.1 Foundational Concepts...... 1 1.2 Purpose...... 4

2 Theory 6 2.1 Spallation Neutron Production and Transport...... 6 2.2 Foils...... 9 2.2.1 Multiple Foil Activation Technique...... 9 2.2.2 Foil Criteria...... 11 2.2.3 Activity During Irradiation...... 13 2.2.4 Activity After Irradiation...... 15 2.2.5 Decays and Counts During the Counting Time...... 16 2.3 Gamma Spectrometry...... 18 2.3.1 Gamma-ray Physics...... 18 2.3.2 The High Purity Germanium (HPGe) Detector...... 21 2.3.3 The Gamma-ray Spectrum...... 22 2.3.4 Counting Statistics...... 25 2.4 Spectral Unfolding...... 27 2.4.1 Response Matrix...... 27 2.4.2 Monte Carlo Neutron Transport Methods...... 30

3 Experimental Apparatus 32 3.1 The Spallation Neutron Source (SNS)...... 32

vi 3.1.1 The Target Station...... 32 3.1.2 Neutron Beamlines...... 34 3.1.3 Neutron Instruments...... 35 3.2 Foil Selection...... 36 3.3 Gamma Spectrometer...... 38 3.3.1 Ortec PopTop...... 38 3.3.2 Electronics and Software...... 39 3.3.3 Graded Shield...... 40 3.3.4 Liquid Nitrogen LN2 Dewars...... 41 3.3.5 Calibration Sources...... 41 3.3.6 Sample Holder...... 42

4 Computational Tools 44 4.1 Nuclear Data...... 44 4.1.1 ENDF...... 44 4.1.2 LA150...... 46 4.1.3 LAHET in Monte Carlo N-Particle eXtended (MCNPX)...... 47 4.1.4 TALYS-based Evaluated Nuclear Data Library (TENDL)-2010... 47 4.1.5 Other Cross Section Data...... 48 4.1.6 HILO2k...... 48 4.2 Software...... 50 4.2.1 Monte Carlo N-Particle eXtended (MCNPX)...... 50 4.2.2 MAXED...... 51 4.2.3 GRAVEL...... 56 4.2.4 Integral Quantities Utility (IQU)...... 56 4.2.5 Other Unfolding Methods...... 58 4.2.6 Custom Programs and Additional Software...... 59 4.3 Application of Software to Speciﬁc Problems...... 61 4.3.1 Estimating of Foil Activity Using MCNPX and Other Tools..... 61 4.3.2 Calculation of Response Matrix Elements Using MCNPX...... 62 4.3.3 Calculation of Gamma-ray Self-Shielding Using MCNPX...... 65

vii 4.3.4 Correcting for Time-Varying Flux...... 65 4.4 Other Corrections...... 67 4.4.1 Partially Irradiated Foils...... 67 4.4.2 Calibration Source Size Versus Foil Size...... 68

5 Measurement Results 69 5.1 July 8, 2009 ARCS...... 69 5.1.1 Foil Irradiation...... 69 5.1.2 Gamma Spectrometry...... 73 5.1.3 Unfolding...... 83 5.2 June 22, 2011 POWGEN...... 94 5.2.1 Foil Irradiation...... 94 5.2.2 Autoradiography...... 95 5.2.3 Gamma Spectrometry...... 97 5.2.4 Unfolding...... 101

6 Conclusions 102 6.1 ARCS Measurements Conclusions...... 102 6.2 POWGEN Measurements Conclusions...... 103 6.3 Future Work...... 105 6.4 General Conclusions...... 108

Bibliography 109

A Units and Constants 118

B Experimental Foils 119

C Foil Gamma-ray Self-Shielding Factors 121

D Reaction Cross Section Plots 131

Vita 141

viii List of Tables

4.1 HILO2k neutron energy groups from 247.2 keV to 2 GeV...... 49

5.1 Power history corrections for the (Wide) Angular-Range Chopper Spectrometer (ARCS) foil irradiation. Corrections are deﬁned by equation 4.41...... 71 5.2 Candidate peaks in the gamma-ray spectrum of the ARCS foil irradiation.. 76 5.3 Candidate peaks...... 99 5.4 Expected peaks...... 100

B.1 Foils irradiated at ARCS and POWGEN. Where multiple mass values are listed, each value indicates a single foil...... 120

C.1 Gamma-ray self-shielding factors for the aluminum foil listed in Table B.1. Error bars represent absolute standard error...... 122 C.2 Gamma-ray self-shielding factors for the bismuth foil listed in Table B.1. Error bars represent absolute standard error...... 123 C.3 Gamma-ray self-shielding factors for the cobalt foil listed in Table B.1. Error bars represent absolute standard error...... 124 C.4 Gamma-ray self-shielding factors for the copper foil listed in Table B.1. Error bars represent absolute standard error...... 125 C.5 Gamma-ray self-shielding factors for the gold foil listed in Table B.1. Error bars represent absolute standard error...... 126 C.6 Gamma-ray self-shielding factors for the indium foil listed in Table B.1. Error bars represent absolute standard error...... 127

ix C.7 Gamma-ray self-shielding factors for the iron foil listed in Table B.1. Error bars represent absolute standard error...... 128 C.8 Gamma-ray self-shielding factors for the nickel foil listed in Table B.1. Error bars represent absolute standard error...... 129 C.9 Gamma-ray self-shielding factors for the titanium foil listed in Table B.1. Error bars represent absolute standard error...... 130

x List of Figures

2.1 Illustration of the spallation process showing the intranuclear cascade followed by evaporation and high-energy ﬁssion of highly excited nuclei [6]...7 2.2 Calculated spallation neutron spectrum (before moderation) from a tungsten target bombarded by 800 MeV protons compared to a typical neutron spectrum from thermal neutron ﬁssion of 235U. Note the spallation spectrum extends up to nearly the incident particle energy [6]...... 8 2.3 Reaction cross sections for several 209Bi(n,xn) threshold reactions. See Ap- pendixD for additional threshold reaction cross sections...... 10 2.4 Behavior of activity during and after irradiation. After irradiation, the foil activity is measured during the counting time interval. During the short cooling interval, the activity decays but counting does not occur...... 16 2.5 The gamma-ray reaction cross sections of germanium [9] illustrate the three primary mechanisms of gamma-ray interaction. Note the K shell absorption edge at 11.1 keV and the threshold of pair production at 1022 keV...... 19 2.6 The gamma-ray spectrum of 28Al illustrating the major features [10]..... 23

3.1 The SNS target station [17]...... 33 3.2 Diagram of the Ortec PopTop detector and liquid nitrogen (LN2) dewar [41]. This illustration shows the detector mounted horizontally at a 90◦ angle, while the actual detector was mounted vertically...... 39 3.3 Graded lead shield with copper liner above the 30 L LN2 dewar...... 40 3.4 Calibration source used for spectrometer energy and eﬃciency calibrations. 42 3.5 Exploded view of plastic sample holder designed and fabricated for this work. 43

xi 4.1 Reaction cross section data for 27Al. Note the discontinuities at 20 MeV for the ENDF/B-VII.0 data, which are not present in the TENDL-2010 data.. 45 4.2 Dimensions of the sources used for calibration of the gamma-ray spectrometer [72]...... 68

5.1 Bismuth, copper, nickel, and aluminum foils along with aluminum holder as modeled in MCNPX and visualized with the VisEd (left) and Sabrina (right) software...... 70 5.2 Beam power during the July 8, 2009 ARCS foil irradiation...... 71 5.3 Radiation control technician checks the dose rate on the foils as they are removed the sample position after irradiation...... 72 5.4 Foil activity was measured on the detector end cap. A white index card was used to reduce possible contamination of the detector...... 73 5.5 Energy calibration curve for the HPGe spectrometer used for the ARCS foil irradiation...... 74 5.6 Eﬃciency calibration curve for the HPGe spectrometer used for the ARCS foil irradiation...... 75 5.7 The gamma-ray spectrum of the foils irradiated at the ARCS beamline... 77 5.8 Time analysis measuring the decay constant for the 202Bi and 203Bi candidate peaks...... 79 5.9 Time analysis measuring the decay constant for the 204Bi and 205Bi candidate peaks...... 80 5.10 Time analysis measuring the decay constant for the 206Bi candidate peaks. 81 5.11 Time analysis measuring the decay constant for the 58Co and 24Na candidate peaks...... 82 5.12 Calculated ARCS group source spectrum...... 83 5.13 Calculated ARCS ﬂux spectrum at the sample location...... 84 5.14 The ratio of calculated counts to measured counts for the 202Bi and 203Bi candidate peaks...... 85 5.15 The ratio of calculated counts to measured counts for the 204Bi and 205Bi candidate peaks...... 86

xii 5.16 The ratio of calculated counts to measured counts for the 206Bi candidate peaks...... 87 5.17 The ratio of calculated counts to measured counts for the 58Co and 24Na candidate peaks...... 88 5.18 Flux-to-dose-rate conversion factor as a function of energy used to compute the dose rate metric...... 89 5.19 The unfolded reaction rate to measured reaction rate ratio after initial unfolding and the initial unfolded spectrum as compared to the calculated spectrum. The calculated spectrum is scaled by an overall factor of 0.32 for the best fit to the measured data...... 91 5.20 The unfolded reaction rate to measured reaction rate ratio after reanalyzed unfolding and the unfolded spectrum as compared to the calculated spectrum. The calculated spectrum is scaled by an overall factor of 0.32 for the best fit to the measured data...... 93 5.21 Beam power during the June 22, 2011 POWGEN foil irradiation...... 94 5.22 Autoradiograph for the foils irradiated at the POWGEN beamline...... 96 5.23 Energy calibration curve for the HPGe spectrometer used for the POWGEN foil irradiation...... 97 5.24 Efficiency calibration curve for the HPGe spectrometer used for the POW- GEN foil irradiation...... 98 5.25 The POWGEN gamma spectrum...... 100 5.26 The POWGEN neutron source spectrum...... 101

6.1 Exploded side view of a foil packet designed to address issues encountered with the POWGEN foil irradiation...... 106

xiii C.4 Gamma-ray self-shielding factors for the copper foil listed in Table B.1. Error bars represent absolute standard error...... 125 C.5 Gamma-ray self-shielding factors for the gold foil listed in Table B.1. Error bars represent absolute standard error...... 126 C.6 Gamma-ray self-shielding factors for the indium foil listed in Table B.1. Error bars represent absolute standard error...... 127 C.7 Gamma-ray self-shielding factors for the iron foil listed in Table B.1. Error bars represent absolute standard error...... 128 C.8 Gamma-ray self-shielding factors for the nickel foil listed in Table B.1. Error bars represent absolute standard error...... 129 C.9 Gamma-ray self-shielding factors for the titanium foil listed in Table B.1. Error bars represent absolute standard error...... 130

D.1 Threshold reaction cross sections from the TENDL-2010 library for aluminum.132 D.2 Threshold reaction cross sections from the TENDL-2010 library for bismuth. 133 D.3 Threshold reaction cross sections from the TENDL-2010 library for cobalt. 134 D.4 Threshold reaction cross sections from the TENDL-2010 library for copper. 135 D.5 Threshold reaction cross sections from the TENDL-2010 library for gold... 136 D.6 Threshold reaction cross sections from the TENDL-2010 library for indium. 137 D.7 Threshold reaction cross sections from the TENDL-2010 library for iron... 138 D.8 Threshold reaction cross sections from the TENDL-2010 library for nickel.. 139 D.9 Threshold reaction cross sections from the TENDL-2010 library for titanium.140

xiv List of Abbreviations

ASCII American Standard Code for Information Interchange

ASTM American Society for Testing and Materials International

ARCS (Wide) Angular-Range Chopper Spectrometer

CDF cumulative distribution function

DIM Detector Interface Module

EAF European Activation File

ENDF Evaluated Nuclear Data File

FWHM full-width half-max

HPGe High Purity Germanium

HV high voltage

IQU Integral Quantities Utility

JENDL Japanese Evaluated Nuclear Data Library

LN2 liquid nitrogen

MCA Multi-Chanel Analyzer

MCNPX Monte Carlo N-Particle eXtended

ORNL Oak Ridge National Laboratory

xv RSE Relative Standard Error

SA Simulated Annealing

SAC Sample Activation Calculator

SciPy Scientiﬁc Python

SNS Spallation Neutron Source

TENDL TALYS-based Evaluated Nuclear Data Library

TCS True Coincidence Summing

UMG Unfolding with Maxed and Gravel

xvi Chapter 1

Introduction

1.1 Foundational Concepts

The Spallation Neutron Source (SNS) at Oak Ridge National Laboratory (ORNL) produces neutron beams for scientiﬁc use. The SNS is an accelerator-driven neutron source that directs an intense high-energy proton beam into a liquid mercury target to produce spallation neutrons. The neutrons pass through cryogenic or ambient temperature moderators and then down one of twenty-four beam transport lines to an instrument area. Materials placed in the sample position of an instrument scatter the incident neutrons into a surrounding detector array. Currently, the SNS has 14 such instruments in operation with 5 more under construction and several more planned. Each neutron instrument has been designed using a unique combination of beamline components and detectors to measure various material properties by collecting data from the scattered neutrons. Neutrons in the cold and thermal energy regimes (∼1 meV–2 eV) are the most useful to the scattering experiments, but higher energy neutrons (up to ∼1 GeV) are also present in the beams. Neutrons in several energy ranges, including thermal and high-energy, can be measured through foil activation techniques. Foil activation techniques involve the insertion of material probes into a neutron radiation ﬁeld for the purpose of characterizing the radiation’s energy distribution (spectrum) or intensity. Neutrons interact with the foil causing its constituent nuclei to undergo nuclear transmutations producing product nuclides. Some of these newly created

1 products will be unstable against nuclear decay and spontaneously disintegrate over time, often producing gamma radiation. The number of product nuclei is a function of the foil’s propensity for a nuclear reaction (reaction cross section) and the speed-weighted energy distribution of the incident neutron radiation (neutron scalar flux spectrum). If the reaction cross section and foil composition are well characterized, and the number of product nuclei are measured, unknown properties of the incident neutron flux can be deduced. Both the reaction cross section and the neutron energy spectrum are functions of the incident neutron energy. Unless the reaction cross section exhibits an unambiguous energy feature like a precipitous decrease or increase in magnitude at some energy, the energy dependence of the neutron flux cannot be extracted from the measurement. To reveal the energy dependence of the flux, several foils of differing material can be used, each foil having one or more reactions with distinct responses over ranges of energies. This is the multiple foil activation technique, and it uses threshold reactions, those with a zero cross section value below a threshold energy, to accomplish the required energy discrimination. To measure the quantity of radionuclides produced in the foil, the decay gamma radiation can be used. Gamma radiation can carry away the energy due to a nucleus’s transition between discrete nuclear energy levels during the decay process. The structure of the nuclear levels is typically complex, but it is also unique to each nuclide. The relative proportion of the discrete energies of the gamma-rays is contained in the gamma- ray spectrum, and measured spectrometric data have long been catalogued along with their parent nuclides. Using this catalogued data, a measured gamma-ray spectrum acts as a signature that identifies the decaying nuclide in an activated foil. While the exact energies and relative abundance of the resulting gamma-rays identify the decaying nuclide, the total gamma-ray intensity quantifies the number of decays present in an activated foil during a particular time interval. The energy and intensity of gamma radiation is measured with a gamma-ray spectrometer. While several varieties exist, the HPGe gamma-ray spectrometer is the standard instrument due to its high energy resolution. When a gamma-ray interacts with the atoms in the spectrometer’s single crystal of germanium, it transfers some or all of its energy to the atomic electrons via one of several possible physical mechanisms. The

2 spectrometer records the number of displaced electrons and holes due to the incident gamma radiation during a brief (µs) time interval. The number of displaced electrons and holes produced is proportional to the total energy imparted by the incident gamma photon. If all of the gamma photon’s energy is deposited in the germanium, the number of displaced holes and electrons is proportional to the total energy of the incident gamma photon. The activity is defined as the magnitude of the nuclear decay rate. For gamma- ray emitting decays, the activity is proportional to the rate of gamma radiation emission. While measuring a single gamma-ray’s energy may only require a single event in the spectrometer, activity measurements require measuring many gamma-rays over a time interval. Because the process of radioactive decay is a random occurrence, measuring more gamma- rays over a longer period of time reduces the statistical uncertainty associated with the activity measurement. Once the activity due to a particular product nuclide has been measured, the neutron flux spectrum that induced the activity remains to be determined. Given the foils’ material properties, if the neutron flux spectrum is known, the resulting activity can be computed easily. The converse situation, computing the neutron flux spectrum from the resulting activity given the foils’ material properties, is more difficult. Generally, when a model and its output are known, but the model input is unknown, the problem is called an inverse problem. To determine the neutron flux spectrum from foil activity measurements is to an solve inverse problem through a process known as unfolding. Mathematically, unfolding is deconvolution, the process of distinguishing the contributing components of a combined function. Because a finite number of measurements are used in the unfolding, a continuous neutron flux spectrum cannot be determined, rather the flux spectrum consists of group flux values over many discrete energy groups. Factors influencing the selection of energy group structure may include the expected behavior of the flux over the entire energy range and the energy detail required in a subsequent calculation. Often these issues dictate that the number of energy groups exceeds the number of activity measurements used to determine the group flux values. When this occurs, solving for the neutron flux is an underdetermined inverse problem which cannot be computed by direct calculation. A specialized iterative algorithm can be employed to find a solution flux spectrum that is consistent with the foil

3 activity measurements. By irradiating multiple foils at the SNS instruments, measuring their induced gamma-ray activity, and then using computational unfolding methods, the high-energy neutron ﬂux spectra at the SNS can be measured.

1.2 Purpose

The purpose of this work has been to develop a foil activation method to measure the high- energy (1–120 MeV) neutron flux spectra at the SNS. This method has been developed by researching the scientific literature, assembling experimental apparatus, performing measurements, analyzing their results, and refining the technique based on experience. Mea- suring the high-energy neutron flux spectrum provides further empirical data to validate the SNS shielding design and sample activation calculations. Consideration of the high-energy neutrons was an important component of the beamline and target monolith shielding design at the SNS. The SNS currently produces neutrons with energies to nearly 1 GeV, and after future upgrades, may produce neutrons with energies to nearly 1.4 GeV. An abundance of data exist for shielding neutrons in the energy range typical of reactors, but considerably less data exist beyond this range. High- energy neutrons produced at the SNS have a much greater penetrating power and require regular concrete1 shielding up to 2m thick [1]. For high-energy neutron transport, the SNS shielding design employed nuclear collision models and multigroup transport libraries derived from nuclear collision models [2]. Nuclear models are less accurate than evaluated cross section data, but relatively few evaluated data exist above 20 MeV. Therefore, standard cross section libraries such as Evaluated Nuclear Data File (ENDF)/B mostly rely on nuclear models above 20 MeV [3]. Because the SNS produces neutrons in this unconventional energy regime, and because radiation shielding is essential for personnel protection, the shielding design required validation with measurements. Many safety and operational measurements were completed during SNS commissioning in 2006, but until this work, neutrons in the high-energy range had not yet been measured. Characterizing the large number of activated user samples at the SNS provides

1The shielding implemented at SNS includes both regular and high-density concrete.

4 another motivation for measuring the high-energy flux spectra. Accurate measurements of post-irradiated samples is difficult and time consuming, so prior to irradiation, SNS staff estimate the expected dose rate and isotopic inventory of each sample by using the Sample Activation Calculator (SAC) developed at the SNS [4]. Should the activated user samples exceed administrative dose-rate or isotope levels, the samples may be retained at the facility indefinitely or until the activity has decayed below release levels. Measurement of the high-energy neutron flux spectra will provide validated flux spectra, and the foil activity measurements serve as an empirical benchmark for the activation code’s estimates. Improved sample activation estimates will reduce the number of samples conservatively retained on site, allowing users greater flexibility and reducing the sample stewardship burden to the facility. Finally, high-energy neutron flux spectra measurements may serve some yet unanticipated purpose. The measurements could serve a diagnostic purpose for beamlines or be incorporated into future beamline shielding models. The utility of these measurements are likely to broaden once they become routinely available. But before they can be used for any of these purposes, a method of measurement must be developed. Developing that method has been the goal of this work.

5 Chapter 2

Theory

This chapter describes the necessary theoretical foundation for developing a method of high-energy neutron ﬂux spectrum measurement at the Spallation Neutron Source (SNS).

2.1 Spallation Neutron Production and Transport

Neutrons can be produced by several means including nuclear reactors, (α,n) sources, and radioactive isotopes (e.g. 252Cf). The SNS generates neutrons through a diﬀerent mechanism than these sources. Nuclear spallation is analogous to the eponymous mechanical spallation process, the ejection of fragments from a object when impacted by another object. Nuclear spallation occurs when energetic (> 100 MeV) subatomic particles (protons, neutrons, pions, etc.) collide with a nucleus to eject subatomic particles. At neutron scattering facilities, spallation is designed to occur within a material target. Unlike (n,xn) reactions, spallation reactions are not considered to form an intermediate compound nucleus. The high energies of the projectile particles means their de Broglie wavelength is short enough to interact with individual nucleons inside the nucleus. The projectile particle undergoes a series of direct reactions with the nucleons to create an “intranuclear cascade” whereby several nucleons may be ejected and the nucleus is left in an energet- ically excited state. The primary projectile and the secondary spallation particles may have enough kinetic energy to induce further spallation reactions in other nuclei. The

6 Figure 2.1: Illustration of the spallation process showing the intranuclear cascade followed by evaporation and high-energy ﬁssion of highly excited nuclei [6]. highly-excited nucleus relaxes to its ground state by subsequently “evaporating” nucleons, primarily neutrons. In heavy nuclei such as tantalum, tungsten, and lead, high-energy ﬁssion also competes with nuclear evaporation in the excited nucleus [5] (Figure 2.1). The spallation and evaporation neutrons created during and after the collision span the range of energies up to nearly1 the incident particle energy (Figure 2.2). Be- fore most of these neutrons can be useful to neutron scattering experiments, they must be cooled to cold and thermal neutron energies. This is accomplished with a dedicated moderator, typically composed of hydrogenous material, which reduces the mean neutron energy through scattering collisions. An ideal gas of neutrons at thermal equilibrium have a Maxwellian distribution of energies E

√ 2π − E M(E) = Ee kT (2.1) (πkT )3/2

1The nuclear binding energy is typically a small fraction of the incident particle energy.

7 Figure 2.2: Calculated spallation neutron spectrum (before moderation) from a tungsten target bombarded by 800 MeV protons compared to a typical neutron spectrum from thermal neutron fission of 235U. Note the spallation spectrum extends up to nearly the incident particle energy [6]. at temperatures T , where k is the Boltzmann constant. Although neutrons transiting the moderator will not be precisely in thermal equilibrium, a Maxwellian distribution of energies is usually a reasonable approximation for neutron energies near kT , where T is the moderator temperature. Neutrons leaving the target and the moderator travel in a distribution of directions. Neutronically reflecting material may surround the target to increase the probability that errant neutrons will enter the moderator. Liners, decouplers, and other forms of neutron poisons also may be used to alter the time and energy characteristics of neutrons used in scattering experiments. Once neutrons have been quasi-thermalized in the moderator, they propagate through surrounding material such as shielding. The neutron beamlines are essentially pathways in the shielding through which neutrons can travel virtually unimpeded. Most beamlines at scattering facilities make use of supermirror neutron guides. These guides propagate wavelike low-energy neutrons with small incident scattering angles by reflec-

8 tion. Neutron guides can increase the low-energy neutron flux delivered to the end of the transport line by a few orders of magnitude. Higher energy neutrons are unaffected by neutron guides, so slightly curving a guide along a beamline filters the high-energy component. Unguided neutron flux decreases over the length of the beamline with the typical 1/r2 relationship, where r is the distance from a point source. Because this work is concerned with high-energy neutrons, guide propagation of low-energy neutrons will be of little consequence, but the mechanism is mentioned here for completeness. The transport of non-wavelike neutrons in non-multiplying media with a time-

independent external source qext is described by this form of the Boltzmann neutron transport equation:

Ωˆ · ∇~ + Σt(~r, E) ψ(~r, Ωˆ,E) = qext(~r, Ωˆ,E) Z ∞ Z 0 0 0 0 + dE dΩ Σs(~r, Ωˆ · Ωˆ,E → E)ψ(~r, Ωˆ,E) (2.2) 0 4π

where ψ(~r, Ωˆ,E) is the angular ﬂux at position ~r traveling in direction Ωˆ of energy E,Σt is the macroscopic total cross section, and Σs is the macroscopic scattering cross section from direction Ω0 to Ω and from energy E0 to E. The spallation process of neutron production will be considered an external source, and the materials used are non-multiplying, so equation 2.2 is suﬃcient to describe most neutron transport in this work.

2.2 Foils

2.2.1 Multiple Foil Activation Technique

Foil activation can be used as a measurement of neutron flux and flux spectra over a wide range of energies. Some materials exhibit large reaction cross sections in the thermal energy range (e.g. 197Au(n,γ)198Au), and others have large reaction cross section resonance peaks in the epithermal range (e.g. 115In(n,γ)116mIn ). Reactions of these types have cross section values typically ranging from 10–104 b. The energy features of these reaction cross sections allow the reaction products to be mostly attributed to neutrons in those energy ranges. Thus, measurement of the reaction products in the foils can be related to the neutron flux in these energy ranges.

9 Figure 2.3: Reaction cross sections for several 209Bi(n,xn) threshold reactions. See Ap- pendixD for additional threshold reaction cross sections.

The same principle applies to measuring fast and high-energy neutrons (E >∼ 0.5 MeV) with foil activation, but the reaction cross sections have a different energy feature. Inelastic scattering, (n,xn), and some (n,p) and (n,α) reactions only occur when the incident neutrons exceed a threshold energy [7]. These threshold reactions have effectively a zero cross section below the threshold, but some finite value above the threshold (Figure 2.3). Therefore, activation from these reactions are only due to neutrons with energies above the threshold. Threshold reactions have cross section values typically in the range of 10−3–10 b.

When the activation from multiple reactions with diﬀering thresholds are measured, determining the neutron ﬂux spectra with greater energy detail is possible. Multi- ple threshold reactions may occur with the same nuclide (e.g. 58Ni(n,p) and 58Ni(n,2n)), but typically more reactions can be included by irradiating multiple foils. For example, 24Na nuclei from 24Mg(n,p) reactions are due to neutrons with energies greater than the

10 threshold of 6.0 MeV, and nuclei 57Ni from 58Ni(n,2n) reactions are due to neutrons with energies greater than the threshold of 13.0 MeV [8]. The neutron ﬂux in the energy ranges 6.0 MeV< E <13.0 MeV and E >13.0 MeV can be determined by measuring the reaction activation. When several reaction thresholds span a broad range of energies, a discrete neutron ﬂux spectra in that range can be determined.

Advantages of the multiple foil activation technique:

1. Because of the relatively low value of the reaction cross sections, foils can be several millimeters thick without disturbing the neutron ﬁeld.

2. The foils are practically insensitive to gamma radiation, so they may be employed in mixed radiation ﬁelds.

3. Foils have a small proﬁle and can ﬁt in many locations.

4. Foils can be placed in high radiation environment that would damage or degrade other kinds of instrumentation.

Disadvantages of the multiple foil activation technique:

1. Because foil activity is measured after irradiation, live measurements are not possible.

2. Activity measurements, and therefore counting statistics, depend on the time between when irradiation ends and counting begins, so measurements are time sensitive.

3. Well-known cross sections and foil material properties are required.

2.2.2 Foil Criteria

Foils selected for irradiation and subsequent gamma spectrometry should possess several useful qualities.

• The foil’s composition should be known precisely because unknown contaminants will introduce errors in mass and possibly activity. Foils of non-uniform composition will complicate the activation analysis and should be avoided.

11 • Cross section values for selected reactions should be well known over the incident neutron energy range.

• Preferably, the half-life of product nuclides used for measurements range between several hours and a few years, but those with half-lives outside this range also may be useful. Nuclides with short half-lives will achieve equilibrium faster during irradiation, but will also decay more quickly after irradiation. For short lived nuclides, much of the induced activity may decay during the time interval between irradiation and gamma spectrometry. For long lived nuclides, an insigniﬁcant number of decays may occur during gamma spectrometry, leading to poor counting statistics.

• For a selected reaction, the value of the cross-section should be large enough over a range of energies to produce signiﬁcant activity during the counting time.

• Competing reaction channels should be minimized. This may be achieved in part by selecting foils composed of few nuclides, preferably only one. Foils composed of a naturally monoisotopic element make good candidates. The (n,xn) reactions make good candidate reactions because the reaction products cannot be produced through competing reaction channels. Inelastic scattering reactions (n,n’) may be used, provided the reaction cross sections are known.

• Products must emit gamma radiation during decay, and the gamma’s energy must be considered. The likelihood of detecting a photon of a particular energy is affected by the foil self-attenuation and the detector efficiency. Harder gammas are more likely to escape the foil, but the High Purity Germanium (HPGe) detector efficiency is maximal around 130 keV.

• A decay channel with greater gamma emission probability will improve counting statistics, all other factors being equal.

• The foil surface area should either be large enough to fully encompass the beam, or should be small enough to be fully immersed in the beam. Non-uniform or partial irradiation of the foil complicates the activation analysis.

12 • The number of foil nuclei should be great enough to produce signiﬁcant activity during counting. The number of nuclei can be increased by increasing the density, thickness, or surface area of the foils.

• The foil should be thin enough that the ﬂux is not signiﬁcantly perturbed as the beam transits it.

• The foil should be thin enough such that the product gamma radiation intensity is not signiﬁcantly attenuated.

2.2.3 Activity During Irradiation

Radioactive decay for any nuclide i occurs with a constant relative rate λi. In any two time intervals of the equal length, the same proportion of decays occur relative to the total number of nuclei Ni. The activity Ai(t) is magnitude of the decay rate:

dN (t) i = − λ N (t) (2.3) dt i i

Ai(t) = λiNi(t) (2.4)

The solution to the diﬀerential equation 2.3 is the familiar exponential-decay equation:

−λit Ni(t) = Ni(0)e (2.5)

During irradiation, the nuclide i can be produced and destroyed by reactions due to a time- independent2 (constant) neutron ﬂux φ(E). Three processes determine the net production rate of the Ni nuclei:

dN (t) i = production − destruction − decay dt Z ∞ Z ∞ dNi(t) = Nj(t) σj(E)φ(E)dE − Ni(t) σi(E)φ(E)dE − (Ni(t)λi) (2.6) dt 0 0

2Activity due to a time-dependent ﬂux will be considered in section 4.3.4

13 The target nuclei Nj have a microscopic reaction cross-section of σj, and the Ni nuclei have a cross-section of σi. With the initial condition Ni(0) = 0 and using

Z ∞ σφ = σ(E)φ(E)dE (2.7) 0

to simplify the notation, equation (2.6) can be solved:

dN (t) i = σ φN (t) − N (t)σ φ − N (t)λ dt j j i i i i dNi(t) e(λi+σiφ)t + (λ + σ φ)N (t)e(λi+σiφ)t = σ φN (t)e(λi+σiφ)t dt i i i j j Z t d Z t (λi+σiφ)t (λi+σiφ)t Ni(t)e dt = σjφNj(t)e dt 0 dt 0 Z t (λi+σiφ)t −σj φt (λi+σiφ)t Ni(t)e = σjφNj(0)e e dt 0 N (0)σ φ j j −σj φt −(λi+σiφ)t Ni(t) = e − e (2.8) λi + σiφ − σjφ

The activity can be expressed by combining equations (2.4) and (2.8):

N (0)σ φ j j −σj φt −(λi+σiφ)t Ai(t) = Ni(t)λi = e − e (2.9) 1 + 1 (σ φ − σ φ) λi i j

Two practical approximations can be made:

1. Destruction of target nuclide j can be neglected σjφ 1 which implies: −σ φt e j ≈ 1 and σiφ − σjφ ≈ σiφ

2. Product nuclei i will decay much faster than they will be transmuted: λi σiφ

These approximations are justified for the cross sections and fluxes under consideration for reasonable irradiation times. For example, using conservative estimates of 10 b for an average threshold reaction cross section and an estimated average high-energy 9 n −14 – 1 flux of 10 cm2s , the microscopic reaction rates are 10 s reactions per atom. This value is much smaller than any decay constant under consideration, and to destroy 1% of the target nuclei would take more than 1012 s (3.2 × 104 years).

14 Using these two approximations, equation (2.8) can be simpliﬁed to the foil activity Aφ(t) during irradiation at time t due to a time independent ﬂux φ(E):

−λit Aφ(t) =Njσjφ(1 − e ) (2.10)

Written with explicit energy dependence as:

Z ∞ −λit Aφ(t) =Nj(1 − e ) σj(E)φ(E)dE (2.11) 0

Equation (2.10) illustrates the asymptotic behavior of the activity as a function of time. −λ t As t → ∞, e i → 0 and the activity approaches the saturation level Asat.

lim Aφ(t) = Njσjφ t→∞

lim Aφ(t) = Asat t→∞

Njσjφ is the reaction rate, so:

Saturation Activity = Reaction Rate (2.12)

Asat = Njσjφ

This is the equilibrium condition, when the rate of decay is equal to the rate of production. As Figure 2.4 illustrates, the activity during irradiation approaches saturation after ∼7– 8 half-lives.

2.2.4 Activity After Irradiation

The activity Ai(t) due to nuclide i after irradiation follows the exponential decay law:

−λit Ai(t) = Ai(t0)e (2.13)

where t0 is any previous time after irradiation and λi is the decay constant. Because activity measurements occur a ﬁnite time after irradiation ends, it is convenient to divide the post-irradiation time into two intervals: cooling time and counting time. Cooling time

15 Figure 2.4: Behavior of activity during and after irradiation. After irradiation, the foil activity is measured during the counting time interval. During the short cooling interval, the activity decays but counting does not occur. is the interval after irradiation ends but before measurement begins. Counting time is the interval during which the activity measurement is occurring. Equation 2.13 describes the activity during both these intervals (Figure 2.4).

2.2.5 Decays and Counts During the Counting Time

The number of nuclear decays Di of nuclide i that occur during a time interval [ta, tb] is the time integral of the activity over the interval:

t −λta −λt Z b Ai(t0) e − e b Di = Ai(t)dt = (2.14) ta λ

If the time interval [ta, tb] is the counting time, equation 2.14 describes the number of nuclear decays that occur during that interval. The number of decays during the counting time determines the number of counts measured by the HPGe detector.

16 If t0 is the time at the end of the irradiation, the number of decays during the counting time can be expressed in terms of the saturation activity or the reaction rate. Combining equations 2.10 and 2.14 yields:

−λ t Njσjφ(1 − e i 0 ) D = e−λta − e−λtb (2.15) i λ

The total number of gamma-rays emitted Γi from nuclide i of energy Eγ during the counting period is the gamma emission probability Ii,γ(Eγ) multiplied by the total number of decays.

Γi(Eγ) = DiIi,γ(Eγ) (2.16)

The number of gamma-rays escaping the foil Gi from nuclide i is the number of gamma-rays 3 emitted Γi multiplied by a geometry dependent self-shielding factor Fself (Eγ)

Gi(Eγ) = Fself (Eγ)Γi(Eγ) (2.17)

The energy-dependent total detector eﬃciency is deﬁned as the the total number of gamma-rays counted C in the detector divided by the total number G of gamma-rays escaping the foil. C(Eγ) (Eγ) ≡ (2.18) G(Eγ)

Combining equations 2.15– 2.18, the number of counts expected in a HPGe detector Cˆi

from nuclide i during a time interval [ta, tb] is:

(Eγ)Fself (Eγ)Ii,γ(Eγ) Cˆ (E ) = N σ φ 1 − e−λit0 e−λta − e−λtb (2.19) i γ λ j j

Due to the random nature of radioactive decay the actual number of counts measured in the detector will vary4 from the value given by equation 2.19.

3The self-shielding factor is computed in section 4.3.3. 4The variance in the number of counts is discussed in Section 2.3.4.

17 2.3 Gamma Spectrometry

2.3.1 Gamma-ray Physics

Nuclear gamma-rays are electromagnetic radiation in the energy range of ∼10 keV–10 MeV. This range overlaps with X-rays (∼120 eV–120 keV), and the distinction between the two lies only in their origin. If radiation in this overlapping energy range originates from electrons, it is considered an X-ray, but if the radiation originates from the nucleus, it is considered a gamma-ray. Gamma-rays can interact with both the nucleus and the atomic electrons of an atom, but when considering interactions between the gamma-rays and detector in HPGe spectrometry, the photo-atomic interactions are most relevant. A gamma-ray undergoes one of three primary physical processes when it interacts with an atom: photoelectric absorption, Compton scattering, or pair production.

Photoelectric Absorption

Photoelectric absorption occurs when a gamma-ray is absorbed by an atomic electron, and that electron is ejected from the atomic shell with kinetic energy from the gamma-ray. The kinetic energy of the electron Ee is given by the conservation of energy as:

Ee = Eγ − Eb (2.20)

where Eγ is the gamma-ray energy, and Eb is the energy binding the atomic electron to its shell. Gamma-rays have sufficient energy to eject electrons of the inner K, L, and M atomic shells. Sharp absorption edges exist the in the photo-atomic cross sections (Figure 2.5) where energies less than the edge are insufficient to eject electrons from the next inner shell, while energies above the edge are sufficient. After ejection of the electron, the atom is left in an excited state with excess energy Eb. To release the excess energy, the atom de-excites in one of two ways: the ejection of additional (Auger) electrons, or the redistribution of electron in their shells. When electrons are redistributed, less tightly bound electrons in outer shells fall into the inner shell vacancies, releasing energy through X-ray fluorescence.

18 Figure 2.5: The gamma-ray reaction cross sections of germanium [9] illustrate the three primary mechanisms of gamma-ray interaction. Note the K shell absorption edge at 11.1 keV and the threshold of pair production at 1022 keV.

Compton Scattering

When a gamma-ray imparts a fraction of its energy to an atomic electron, the interaction is called Compton scattering. Compton scattering often occurs with loosely bound electrons where Eγ Eb, so the binding energy can neglected and the electron can be considered free. For the case of a free electron, the kinetic energy imparted to the electron can be derived through conservation of energy and momentum:

0 Conservation of Energy: Ee = Eγ − Eγ (2.21) 1 Conservation of Momentum: Ee = Eγ 1 − 2 (2.22) 1 + Eγ(1 − cosθ)/mec

0 2 where Eγ is the energy of the scattered gamma-ray, θ is the scattering angle, and mec is the rest energy of the electron (∼ 511 kev). By deﬁnition, the full energy of the gamma-ray is not transferred to the electron

19 in Compton scattering, but kinematics prohibit a range of energy loss less than the full energy. From equation 2.21, the energy lost by the gamma-ray is maximal when Ee is

maximal, and from equation 2.22, Ee is maximal when θ = ±π. This is the backscattering case where:

1 Backscattering: Ee = Eγ − 1 2 (2.23) + 2 Eγ mec ! 2 1 mec lim 1 2 = Eγ →∞ + 2 2 Eγ mec m c2 Therefore: E < E − e (2.24) e γ 2

The actual upper limit of energy loss is determined by equation 2.23, and equation 2.24 shows that energy loss will never be within 255.5 keV of Eγ. This prohibited range of energy loss is important to understanding the features of the measured gamma-ray spectrum (Section 2.3.3).

Pair Production

When a particle and its antiparticle collide, the two annihilate in a burst of gamma radiation with equivalent energy. Pair production is the symmetric process to particle annihilation. Pair production occurs within the Coulomb ﬁeld of the nucleus when a gamma-ray transforms into a particle and its antiparticle. The case that is relevant for gamma-ray spectrometry is electron-positron pair production. In order for electron-positron production to occur, the gamma-ray must have 2 at least as much energy as the rest energies of the pair 2mec = 1022 keV. Gamma-rays with less than 1022 keV cannot undergo pair production (Figure 2.5). The excess energy during pair production is distributed equally among the particle pair:

E − 2m c2 E = E = γ e (2.25) e− e+ 2 where Ee− and Ee+ are the electron and positron energies respectively. Once created, the electron and positron continue to interact with surrounding atoms and nuclei. In- evitably, the positron loses its kinetic energy and collides with an electron, thus releasing

20 two annihilation gamma-rays of 511 keV.

2.3.2 The High Purity Germanium (HPGe) Detector

The principle underlying all gamma-ray spectrometers is that the number of ionizations due to the gamma-ray that occur within the detector material is proportional to the energy absorbed. If all of the gamma-ray energy is absorbed in the detector, the number of ionizations is proportional to the energy of the gamma-ray. Gamma-ray spectrometers vary in material, which determines the energy per ionization, and the mechanism used to measure the ionization. Scintillator-based spectrometers use photons to measure ionization, and semiconductor-based spectrometers use electron-hole pairs. The HPGe gamma-ray spectrometer is a semiconductor-based detector. In solid materials, electron energy levels become energy bands. The uppermost occupied energy band, which is responsible for chemical bonding, is known as the valence band. Beyond the valence band, separated (in semiconductors) by a range of prohibited energies called the band gap, lies a conduction band that allows charge to move through the material. In semiconductors, the band gap is small enough that some thermally excited electrons may move from the valence band to the conduction band, allowing some degree of conductivity. When a gamma-ray enters the germanium, it undergoes a series of photoelectric, Compton scattering, and pair production interactions that ultimately cause the promotion of electrons from the valence band into the conduction band. The absence of an electron in the valence band is called a hole. When an electron is promoted to the conduction band, adjacent valence electrons move to ﬁll the vacancy and a hole eﬀectively propagates in the opposite direction. When an electric potential (voltage) is applied across the germanium, the conduction band electrons propagate toward the cathode and holes propagate toward the anode. By measuring the change in voltage, the number of ionizations, and therefore the absorbed energy, can be measured. Voltage pulses are recorded by an electronics system that includes a Multi-Chanel Analyzer (MCA), which counts and sorts the pulses by height. To distinguish one gamma-ray’s energy from the next, the electrons and holes must propagate through the material quickly. Dislocations and impurities in the semiconductor lattice can reduce the mobility of the charge carriers in the material, so a single

21 crystal of high purity is used. Germanium has become the standard material of semiconductor gamma-ray spectrometers because of the high mobility of its charge carriers and its large absorption coeﬃcient, which allows for detection of higher energy gamma-rays. Un- fortunately, the band gap in germanium is so small that too many electrons move into the conduction band due to mere thermal excitation, rather than promotion due to gamma- ray energy. This “leakage current” is reduced by reducing the thermal excitation of the electrons (cooling the material). For this reason, HPGe systems are operated at liquid nitrogen (LN2) temperatures (77K).

2.3.3 The Gamma-ray Spectrum

In a hypothetical inﬁnitely large detector, every entering gamma-ray would be unable to exit, so each would transfer all of its energy to the detector material. Ionizations would occur and the voltage pulse heights would be recorded by the MCA. Even in this hypothetical scenario, every voltage pulse would not be recorded identically for gamma- rays of the same energy. Uncertainties in the process cause pulses to be recorded with some variance around a central peak. These uncertainties can be added in quadrature to yield an overall energy measurement uncertainty ω:

2 2 2 2 2 ω = ωI + ωP + ωC + ωE (2.26)

2 The intrinsic variance ωI in the actual gamma-ray energy emitted by a nuclide 2 is quite small and can be neglected. The variance in charge production ωP is an inherent property of the detector material so it cannot be reduced. The variance due to charge 2 2 collection ωC is due to crystal defects and geometry. Variance due to electronic noise ωE can be minimized with properly adjusted electronic components of high-quality and a clean supply of power. The total uncertainty in gamma-ray energy measurement is known as resolution. A spectrometer’s resolution, a function of the energy absorbed, is the reason gamma-ray peaks have a ﬁnite width. The peak located at the energy of the incident gamma-ray is known as the full- energy peak or photopeak. When a gamma-ray enters the detector and is absorbed through a photoelectric interaction, the energy of the ejected electron will ultimately lead to the

22 Figure 2.6: The gamma-ray spectrum of 28Al illustrating the major features [10]. promotion of a proportionate number of electrons to the conduction band. When this occurs, the MCA records a count in channels around the full-energy. Real detectors are not inﬁnitely large, so gamma-rays may exit the detector before fully depositing their energy. For this reason, gamma-ray spectra contain several features in addition to the full-energy peak (Figure 2.6). A gamma-ray that Compton scatters once and then exits the detector transfers some of its energy to a Compton electron. Like the photoelectron, the Compton electron causes an energy proportional number of electrons to be promoted to the conduction band. The energy transferred to the Compton electron is governed by equations 2.21 and 2.22 and ranges between:

1 0 < Ec ≤ Eγ − 1 2 (2.27) + 2 Eγ mec

For this reason, the MCA will record these events in channels spanning a continuum of energies less than the full-energy peak. This is known as the Compton continuum, and the upper limit is the Compton edge. Atomic binding energies (neglected in equations 2.21 and 2.22) and spectrometer resolution cause the Compton edge to be softened into a Compton shoulder. One ﬁgure-of-merit of a gamma spectrometer is the peak-to-Compton

23 ratio, which is the ratio of counts in a 60Co full-energy peak to the average counts in the Compton continuum [10]. Should a once-scattered gamma-ray Compton scatter again before exiting the detector, it may deposit more energy than the Compton edge of its ﬁrst scattering. Therefore, the energy range between the Compton shoulder and the full-energy peak is called the multiple Compton region.

For gamma-rays with Eγ > 1022 keV, pair production can occur inside the source material, inside the detector, or in any nearby material. When pair-production occurs in material outside of the detector, the positron quickly loses its kinetic energy and annihilates with an electron producing two ∼511 keV gamma-rays5. These gamma-rays may then enter the detector and deposit their energy. This is one reason an annihilation peak at 511 keV often appears in spectra along with gamma-rays more energetic than 1022 keV. Another source for annihilation gamma-rays is nuclear decay by positron emission, for example β+ 22Na −−→ 22Ne. When pair production occurs within the detector material, two annihilation gamma-rays are also created. Annihilation typically occurs within 1 ns of pair production, which is practically instantaneous compared to the detector charge collection time of 100– 700 ns [10]. One or both of these 511 keV gamma-rays may escape the detector without depositing their energy. When this occurs, two additional peaks appear in the spectrum:

Single Escape Peak: Eescape,1 = Eγ − 511 keV (2.28)

Double Escape Peak: Eescape,2 = Eγ − 1022 keV (2.29)

The full-energy, annihilation, single and double escape peaks along with the Compton continuum form the major features of a gamma-ray spectrum. Additionally, several minor features are also present. A backscattering peak in the range of 200–250 keV may be present [11]. The backscattering peak is due to gamma-rays Compton backscattering oﬀ adjacent material before entering to the detector. Fluorescence X-rays resulting from ejected photoelectrons in the source or adjacent materials also may be detected. Fi- nally, sum peaks resulting from contemporaneous gamma-rays both depositing all of their

5The net momentum during the collision gives rise to small deviations in the energies of the gamma-rays (511 ± δ keV). This causes the annihilation peak in the gamma spectrum to be Doppler broadened.

24 energy also can occur.

2.3.4 Counting Statistics

Counting nuclear decays is a measurement process subject to both the random and systematic errors. Random errors are experimental uncertainties that can be revealed by repeated measurements; those that cannot be revealed by repetition are systematic errors [12]. An important component of the random error of counting measurements is determined by the statistics of radioactive decay. Radioactive decay is a Bernoulli process that occurs with a binomial distribution. The expected number of decays that occur in a time interval [0, t] is the mean µ of the binomial distribution: µ = Np(t) (2.30) which is the number of nuclei N multiplied by the probability of a decay during the time interval, the cumulative distribution function (CDF) p(t) . The variance, or expected squared error, of the binomial distribution is σ2:

σ2 = Np(t)q(t) (2.31) where q(t) is the probability of a non-decay. Using p(t) + q(t) = 1 and equation 2.5:

p(t) = 1 − e−λt (2.32)

q(t) = e−λt (2.33)

When λt 1, q(t) ≈ 1 and the binomial distribution can be approximated very well by a Poisson distribution:

µ ≈ Np(t) (2.34) √ σ ≈ µ (2.35) where the square-root of the variance is the standard deviation σ. To determine the statistics of counting, two ranges of count times are considered: short count times and long

25 count times. In both cases, the error in the time measurement is assumed to be negligible.

Short Count Time

p During a short count time λt 1, Poisson statistics are valid, and σC (t) = C(t) where C(t) is the measured6 number of counts. Assuming a constant background count rate B(t) = Bt˙ , and using equation 2.32 to determine the number of true decays D(t), the count rate C˙ is: d ¨* 1 C˙ = (D(t) + B(t)) = Nλ ¨e−¨λt + B˙ dt

where is the detector eﬃciency. C˙ is constant, therefore C(t) = Ct˙ yields C˙ = C(t)/t.

The standard deviation of the decay rate σC˙ is:

s σ(t) C˙ σ (t) = = (2.36) C˙ t t

The uncertainty in the count rate decreases as the square root of the count time. The net count rate C˙n is the background count rate B˙ subtracted from the total count rate C˙ :

C(t) B(t) C˙n = C˙ − B˙ = − (2.37) t tB where the background counts were independently measured over a time tB. The independent measurements of the background rate and total count rates allows for their uncertainty to be added in quadrature:

σ2 (t) = σ2 (t) + σ2 (t) Cn˙ C˙ B˙ s 2 2 s σC (t) σB(t) C(t) B(t) σCn˙ (t) = 2 + 2 = 2 − 2 (2.38) t tB t tB

The measurement of the net count rate and its uncertainty can be expressed as:

s ˙ C(t) B(t) C(t) B(t) Cn ± σCn˙ (t) = − ± 2 − 2 (2.39) t tB t tB

6The measured number of counts is not necessarily equivalent to the expected number of counts, but it is assumed to be the best estimate [8].

26 Long Count Times

During long count times λt & 1, Poisson statistics would appear not to be a good approximation and use of the binomial distribution would seem necessary. Fortunately, the background count rates remain Poisson distributed, thus the total count rate is Poisson distributed [10]. Therefore, equation 2.39 remains valid for long count times provided C˙ n is regarded as the measured mean net count rate:

C (t) C˙ = n (2.40) n t

To determine the net count rate at the start of counting time, a unit-less correction that accounts for the decay during acquisition can be applied [13]:

λt Decay During Acquisition Corrrection = (2.41) 1 − e−λt

For the net count rate at any time τ ∈ [0, t], the additional factor e−λτ can be applied. The measurement of the net count rate at the start of the counting time and its uncertainty can be expressed as:

s ˙ λt C(t) B(t) C(t) B(t) Cn(0) ± σCn˙ (t) = −λt − ± 2 − 2 (2.42) 1 − e t tB t tB

2.4 Spectral Unfolding

2.4.1 Response Matrix

To compute the neutron flux spectrum from the activity measurements, unfolding calculations are required. In practice, unfolding is carried out by computer software because using one of several specialized iterative algorithms is required. The matrix form of the response equation will be explained in this section, while the particular software and algorithm chosen to find solutions will be discussed in section 4.2.2. The response function R is defined as the ratio of the instrument reading M to the value of the quantity to be measured by the instrument, for a specified type, energy and direction distribution of radiation [14]. In this work, the neutron flux is the desired

27 quantity to be measured, so all response functions are “flux response functions.” Using the above definition, R is the ratio of M to the neutron flux φ:

M R = (2.43) φ

For the purposes of unfolding, M can be any convenient quantity based on the gamma spectrometer reading. For example, M could be the activity or it could be the reaction rate calculated from the activity using equation 2.10.

A single reading Mm is deﬁned as the integral of the energy dependent response function R(E) and the ﬂux φ(E) over an energy range.

Z Emax Mm= R(E)φ(E)dE (2.44) Emin

The energy range between Emin and Emax can be divided into N energy groups with N + 1 group boundaries, and equation (2.44) can be rewritten exactly as:

N X Z En+1 Mm = R(E)φ(E)dE n=1 En N R En+1 R(E)φ(E)dE Z En+1 X En Mm = φ(E)dE R En+1 φ(E)dE En n=1 En with R En+1 R(E)φ(E)dE En Rmn = (2.45) R En+1 φ(E)dE En Z En+1 φn = φ(E)dE (2.46) En N X Mm = Rmn φn (2.47) n=1

Equation (2.45) deﬁnes the N group averaged response function values Rmn for a single

measurement Mm. The N values of φn deﬁned by equation (2.46) are the group ﬂuxes.A total of M measurements are used in the unfolding, so equation (2.47) can be written in a

28 matrix form:

     M1 R11 R12 ...R1N φ1            M2   R21 R22 ...R2N  φ2   =    .   . . .. .  .   .   . . . .  .       MM RM1 RM2 ...RMN φN M~ = R φ~ (2.48)

In equation 2.48, M~ comes from the gamma spectrometer readings, φ~ are solved by using an unfolding algorithm, and the response matrix R must be known. The response matrix elements Rmn are essentially the contribution of the ﬂux φ in each energy group n to the overall spectrometer reading M for a reaction m. The response matrix R could be empirically determined through foil measurements with an energy calibrated neutron source. This empirical technique is possible for neutrons at thermal energies [15], but calibrated high-energy neutron sources are generally unavailable. The response matrix also can be computed based on nuclear cross sections, foil composition, and foil geometry. All response matrices used in this work are computed using the Monte Carlo transport code MCNPX.

To compute the response matrix elements Rmn, the form of the instrument reading M must be specified. For practical reasons, M was selected to be the “reaction rate during irradiation.” MCNPX can output reaction rates, but it does not output activity over time due to radioactive decay. Conversion to activity can only be done in post-processing once MCNPX is finished. Considering equation 2.48, it is simpler to convert the activity measured by the gamma spectrometer into reaction rates rather than converting MCNPX’s reaction rates into activity. Converting activity to reaction rate in the M~ vector entails M operations, while converting reaction rates to activity in the R matrix would entail M × N operations. Having specified the instrument reading M as “reaction rate during irradiation,”

the form of Rmn can be speciﬁed as:

R En+1 N σ (E)φ(E)dE En m m Rmn = (2.49) R En+1 φ(E)dE En

29 2.4.2 Monte Carlo Neutron Transport Methods

The Monte Carlo method, or the method of statistical trials, has long been a useful for neutronics computations. By using basic nuclear data combined with problem specific material geometry and composition, many particle histories are simulated mathematically. Individ- ual histories use pseudo-random numbers selected from natural or modified distributions of position, energy, distance, collision type, and reaction products. The method’s strength is its accuracy, while its weakness is the long computational times required. The subject is far too vast for a detailed description, but some general features are discussed [16]. Monte Carlo methods are used to calculate the sample mean of some quantity, for example, the scalar flux inside some geometrically defined region. The scalar flux can be defined in several equivalent ways, one of which uses the total track length of all particles traversing a region per unit volume per unit time:

1 φ = l (2.50) V where φ is the mean scalar ﬂux, V is the volume of the region, and l is the mean track

length per source particle. The sample meanx ˆN of N histories takes a general form of:

N 1 X xˆ = x (2.51) N N n n=1 where xn is the result of the trial n. A Monte Carlo estimate of the mean scalar ﬂux can be made by using the sample mean of the scalar ﬂux φˆ:

N 1 1 X φˆ = l (2.52) N V N n n=1 where ln is the track length of trial n.

On average, the sample meanx ˆN deviates from the true mean x by some amount. 2 The spread ofx ˆN around x is characterized by the variance of the mean σxNˆ . The variance

30 2 2 of the mean σxNˆ is related to the variance of the distribution of all histories σx:

σ2 σ2 = x xNˆ N σx σxNˆ = √ (2.53) N

Equation 2.53 shows that halving the standard deviation of a Monte Carlo calculation 2 requires four times as many histories. The true variance σx is unknown, but it can be 2 estimated from the sample variance SN :

 2 N N 2 N ! 1 X N X x X xi S2 = (x − xˆ )2 = i − (2.54) N N − 1 n N N − 1  N N  n=1 n=1 n=1

From the Law of Large numbers, it is known that:

lim xˆN = x (2.55) N→∞

From the Central Limit Theorem, it is known that for suﬃciently large N, thex ˆN are normally distributed around x. Therefore, any Monte Carlo estimate and its standard error can be expressed as:

v  2 N u N 2 N ! 1 X u 1 X x X xi xˆ ± S = x ± u i − (2.56) N N N n tN − 1  N N  n=1 n=1 n=1

In this work, through the use of the use of MCNPX software, the Monte Carlo method is used to calculate reaction rates and ﬂuxes inside simulated foil regions.

31 Chapter 3

Experimental Apparatus

This chapter outlines the equipment used to conduct the measurements. The selection of equipment was determined by factors such as practicality, cost, and availability.

3.1 The Spallation Neutron Source (SNS)

3.1.1 The Target Station

The SNS is a particle accelerator facility that directs a high-energy proton beam into a liquid mercury target to generate cold and thermal neutron beams for scientific use. The target station is the collection of systems that surround the target and provide for delivery of neutrons to the beamlines. The target is flowing liquid mercury inside a 316-type stainless steel vessel. The target is approximately 40 cm wide with a height of 10 cm and an active length of 70 cm for neutron production. [18]. A proton pulse of 18 µC approximately 700 ns in length impacts the target at 60 Hz. The proton beam provides up to 1 MW of power to the target. Heat deposited in the mercury is transported through the flowing mercury loop to a heat exchanger. Four dedicated moderators, two above and two below, are adjacent to the target (Figure 3.1). Each moderator serves three or more beamlines. Three of the moderators composed of supercritical hydrogen are maintained at cryogenic temperatures (19 K), while one moderator is composed of light-water at ambient temperature. The moderator param-

32 Figure 3.1: The SNS target station [17]. eters such as location and temperature were determined by the general neutron spectrum requirements of the scattering experiments. While neutrons emerge from the moderators in all directions, the neutron beams are considered to emanate from the “viewed face” of the moderator. The forward moderators have two viewed faces while the rear moderators each have one viewed face [18]. The target and moderators are surrounded by a neutron reflector. Reflectors enhance useful neutron production by returning neutrons to the moderators that either leak from the target to the reflector or escape from the moderators in non-useful directions. But reflectors alter the energies and time structure of the resulting emitted neutrons. Neutron poisons such as gadolinium are used inside and around some moderators for tailoring the spectrum and timing of the neutron beams. “Decouplers” geometrically and neutronically isolate the moderators from the reflectors, while “liners” isolate the moderator viewed surface from the reflectors [5]. Two of the moderators at the SNS target station that employ these neutron poisons are said to be neutronically “decoupled” from their surroundings. The location of the moderators relative to the target precludes spallation neutrons from traveling down beamlines without first scattering in the moderator. This target-

33 moderator configuration is known as “wing geometry.” Wing geometry mitigates the high- energy neutron contamination in the beam at the cost of reducing the overall neutron flux by as much as half when compared to other target-moderator geometries [6]. A core vessel, which serves to contain mercury vapor during target failure, en- cases the target and moderator systems. Extending radially approximately 1–2 m from the viewed face of the moderator are the core vessel inserts, the first beam-shaping components and physical start of the beamlines [19]. Large moveable shutters can be shifted into the flight path of the neutrons to close off individual beamlines while the target continues to produce neutrons. The entire target station is housed in several thousand tons of radiation shielding known as the target monolith.

3.1.2 Neutron Beamlines

The neutron beamlines are pathways leading from the core vessel inserts to to the sample positions of the scattering instruments. Lateral radiation shielding for personnel protection envelops the beamlines. Beamlines may be straight or curved and both use super-mirror guides for the transport of low-energy wavelike neutrons. Some beamlines are so curved that no line- of-sight exists between the viewed face of the moderator and the sample position. As high-energy neutrons are not transported along curved guides, these beamlines are not appropriate for high-energy neutron flux spectra measurements. Pulsed neutron sources like the SNS have an intrinsic neutron beam time structure that is coordinated with the incident proton pulse. Due to the finite time involved in the moderation process, the neutron beam pulses contain a wide spread of energies and need further modification to be useful for specific applications. Choppers, rotating assemblies that alternately block and transmit neutrons with regular frequency, are used to modify the neutron beam. Some beamlines use several choppers over tens of meters for energy selection and neutron pulse time structure. The relative phasing and physical distance of the beamline choppers determines the band of energies transmitted to the sample location. The length of the beamline is important to this work because the high-energy neutron flux will decrease inversely with the square of the distance from the source.

34 The beamlines at the SNS have several varieties of neutron choppers, most of which are not relevant for high-energy neutron ﬂux spectra measurements. Bandwidth (sometimes called “frame-overlap”) choppers set an acceptable band of neutron energies for an experiment and block neutrons from subsequent accelerator pulses. Beamlines may have a T0 chopper, which is speciﬁcally designed to remove the high-energy neutron beam. When high-energy neutron measurements are made on beamlines equipped with T0 choppers, those choppers should be stopped in the open position to transport high-energy beam to the sample location.

3.1.3 Neutron Instruments

(Wide) Angular-Range Chopper Spectrometer (ARCS)

ARCS is an inelastic scattering instrument designed for the thermal and epithermal energy range. Primary science areas for the instrument include the study of lattice dynamics, such as measuring the phonon density of states of diverse materials, and applications to magnetic dynamics in systems from high temperature superconductors to low dimensional quantum magnets [20]. The instrument views the only ambient temperature light water moderator and is equipped with a T0 chopper. The sample position is 13.6 m from the moderator viewed face.

POWGEN

POWGEN is a general purpose neutron diﬀractometer optimized for neutron energies in the range of ∼9–900 meV. Insertable guide sections and the ability to trade resolution for intensity at the analysis stage allow users great latitude to optimize the data range, resolution, and statistical precision for each particular experiment [21]. The instrument views a top upstream cryogenic moderator and is equipped with a T0 chopper. The sample position is ∼ 60 m from the moderator viewed face.

35 3.2 Foil Selection

Foils were selected based upon the general criteria of section 2.2.2, references in the literature, scoping calculations of expected counts, and availability. A set of foils that included bismuth, copper, aluminum, and nickel were immediately available. To generate adequate data for spectrum unfolding at least 6 neutron activation reactions are necessary and about 8–10 are recommended [22]. A list of irradiated foils and their physical characteristics can be found in AppendixB. Bismuth has been acknowledged for its utility in high energy neutron spectrometry at accelerator facilities, and it is the primary foil for measuring the ﬂux spectra above 20 MeV. Measurements of the 209Bi(n,xn), x= 3...12 reaction cross sections [23] have shown good agreement with with the GNASH code based LA150 cross section library [24]. Bis- muth is naturally monoisotopic and its (n,xn) products have usable half-lives ranging from minutes to years. Bismuth has been employed as a threshold activation foil at the ISIS spallation source at Rutherford Appleton Laboratory [25], at the Joint Institute for Nuclear Research in Dubna, Russia [26], and at Brookhaven National Laboratory with experiments on a mercury spallation target [27]. For reactions below 20 MeV (63Cu(n,2n) and 63Cu(n,α)), copper has been often used as an threshold activation detector [22,27,28]. Measurements with copper foils in the fast-energy range is outlined by ASTM Standard Test Method E 523-11 [29]. Copper also has several threshold reactions above 20MeV (65Cu(n,5n) and 65Cu(n,6n)). In the energy range of 40 – 120 MeV, some natural copper (n,xn) reaction cross sections have been measured [30]. Unfortunately, not all of these data are not in agreement with calculated values. No reference was found for the use of copper as a high-energy activation foil, so the usefulness of copper above 20 MeV is possible but questionable. Natural copper is composed of two isotopes (69.17% 63Cu and 30.83% 65Cu) and therefore has competing reactions must be considered. Nickel has two commonly used reactions for activation analysis (58Ni(n,2n) and 58Ni(n,p)). The use of nickel foils for measuring fast neutron reaction rates is outlined in ASTM Standard Test Method E 264-08 [31]. Natural nickel is 68.08% 58Ni and four other isotopes, so competing reactions are an issue. Nickel foils have been used at several

36 facilities for measuring neutron spectra [32–34]. Aluminum is naturally monoisotopic and possesses two usable threshold reactions (27Al(n,α) and 27Al(n,p)). ASTM Standard Test Method E 26611 [35] explains the use of aluminum activation for measuring fast neutron reaction rates. The half-life of 24Na (14.95 hours) and the relatively high reaction cross section of 27Al(n,α) make aluminum an ideal activation detector of fast neutrons. Aluminum foils have been used for this purpose [22, 27, 32]. In addition to the four types of foil that were available immediately, several others were purchased. These included disks of cobalt, gold, indium, iron, and titanium, 1 cm in diameter. Cobalt has reaction thresholds above 20 MeV, which overlap bismuth reaction cross sections. Cobalt is monoisotopic 59Co and the (n,xn) reactions produce products with half-lives in the range of days. Cross section data for these reactions have been measured in the range of 15–40 MeV [36] and in the range of 40–120 MeV [30]. Cobalt foils have been used for high-energy neutron spectrometry [26, 27], and also in the fast- energy range [22, 33, 34]. Because nickel (atomic number Z = 28) and cobalt (Z = 27) are separated by one proton, their reaction products overlap. For this reason, some nickel and cobalt activities should not be measured simultaneously. Gold is the foil material of choice for measuring thermal flux, but it can also be used in the high-energy range. Gold is naturally monoisotopic 197Au and possesses a few (n,xn) reactions with thresholds above 20 MeV. These reactions have been used for high- energy flux spectrum measurements [26]. Gold also has one useable reaction 197Au(n,2n) with a threshold below 20 MeV. There are several reported measurments with gold in this energy range [27, 28, 37]. The indium 115In(n,n’) reaction has an unusually low threshold (0.355 MeV) so it is useful for determining the flux below 1 MeV. Indium is 95.71% 115In, but the other isotope 113In does not produce the 115mIn reaction product, so competing reactions are not a concern. Indium has been used both as an epithermal activation foil [25, 32], and as a threshold reaction foil [34]. Iron has been employed as a fast neutron activation foil [22, 28, 33]. Natural iron is composed of four isotopes, but is 91.75% 56Fe. The 56Fe(n,2n) reaction would be

37 useful, but the product 55Fe is a pure beta emitter so it cannot be detected by gamma spectrometry. Iron has two threshold reactions (54Fe(n,p) and 56Fe(n,p)) with usable half- lives. The use of iron in determining fast neutron reaction rates is guided by the ASTM Standard Test Method E263 - 09 [38]. Titanium foils used for fast measuring neutron reaction rates is outlined by ASTM Standard Test Method E526 - 08 [39], and titanium foils have been used to this purpose [22,28,34]. Three natural isotopes of titanium ( 46Ti, 47Ti, 48Ti) have (n,p) reaction cross sections that cover the range of 2–20 MeV. Cadmium is not used as an activation foil, but its unique neutron absorption behavior makes it useful for activation measurements. Cadmium is an eﬀective absorber of

neutrons below some energy Ec but it passes neutrons of energies above Ec. Ec is known as the “effective cadmium cut-off energy,” and for a 1 mm thick cadmium shield in an isotropic neutron field, it may be taken to be about 0.55 eV. The use of cadmium shielded foils is convenient in separating contributions to the measured activity from thermal and epithermal neutrons. Also, cadmium shielding is helpful in reducing activities due to impurities and the loss of the activated nuclide by thermal-neutron absorption [40].

3.3 Gamma Spectrometer

3.3.1 Ortec PopTop

The gamma-ray detector used in this work is an Ortec coaxial P-type High Purity Germanium (HPGe) GEM25P4 PopTop model. The manufacturer claims the useful range for this model is ∼40 keV–10 MeV, the relative efficiency is ≥ 25%, the peak-to-Compton ratio is 56:1, and the full-width half-max (FWHM) resolution is 0.82 keV at 122 keV and 1.85 keV at 1.33 MeV [42]. The PopTop detector has a 70 mm diameter cylindrical aluminum end cap window of 1 mm thickness, and a bulletized (rounded corners) cylindrical P-type germanium crystal 59 mm in diameter and 54.9 mm in length. The crystal has a 700 µm lithium contact layer on the outside, and a 0.3 µm boron contact layer inside a coaxial bore hole [43]. The PopTop detectors are detachable, and once warmed to room temperature, they can be reconfigured with different liquid nitrogen (LN2) dewars (Figure 3.2).

38 Figure 3.2: Diagram of the Ortec PopTop detector and LN2 dewar [41]. This illustration shows the detector mounted horizontally at a 90◦ angle, while the actual detector was mounted vertically.

3.3.2 Electronics and Software

Many modern gamma-ray spectrometry systems use integrated power electronics and a computer for data acquisition. The electronics used for this work include an Ortec Digi- DART portable digital MCA with DIM-POSGE Detector Interface Module (DIM). The DIM uses a 13-pin power/signal cable to connect to the DigiDART, and a 9-pin pre- ampliﬁer power cable along with BNC coaxial cables to connect to the detector. The BNC cables provide high voltage (HV), pre-ampliﬁer output signal, and a bias shutdown signal. The DigiDART portable MCA interfaces with a computer via a USB cable. By using a rechargeable battery, the DigiDART will continue to read from and provideHV to the detector if the computer crash or loose power. The computer runs the Windows operating system and uses the GammaVision- 32 software. Through GammaVision-32, many electronics settings can be adjusted, includ-

39 Figure 3.3: Graded lead shield with copper liner above the 30 L LN2 dewar. ingHV level, amplifier gain, pulse shaping, and pole-zero cancellation. GammaVision-32 is capable of energy and efficiency calibrations, analysis of spectra including corrections and the use of libraries for identification, and storing spectra in data files for offline analysis. The software also uses a simple programing language for automating repetitive tasks. GammaVision-32 is the primary software used spectral analysis in this work, but custom programs written in the Scientific Python (SciPy) programing language are also used.

3.3.3 Graded Shield

Background radiation from natural sources, such as 40K and radon daughter products, is omnipresent. Radiation from these sources can be reduced by surrounding the gamma-ray detector with a shield, but the shield itself can also act as a source. Lead is often the gamma-ray shielding material of choice. Photoelectric absorption of gamma-rays in lead produces X-ray ﬂuorescence, which can disrupt the gamma-ray spectrum near 80 keV. A few millimeters of copper added to the inside of a lead shield will absorb the lead X-rays.

40 While copper itself produces ﬂuorescence X-rays, these are in the range of 8–9 keV are below the range of interest for gamma spectrometry. The use of layers of materials to shield primary and secondary radiation is known as a graded shield. The graded shield used for this work was produced by a now defunct company called Nuclear Data. The shield consists of a 24 cm diameter cylindrical cavity 30 cm tall with coper lining (3 mm) inside 10 cm of lead with a two piece lead lid of the same thickness. At the base of the shield is a 7.6 cm diameter hole through which the gamma-ray detector penetrates the cavity. The entire shield is on a stand 75 cm high. The LN2 dewar underneath the stand rests on ∼0.6 cm of steel to shield the base of the detector from the concrete ﬂoor (see Figure 3.3).

3.3.4 Liquid Nitrogen LN2 Dewars

Two LN2 dewars were used with this work. The first was an Ortec Gamma Gage cryostat with 7L Multi-Orientation Dewar. The dewar allowed for semi-portable usage of the gamma-ray spectrometer, but the 7 L capacity only allowed for maximum count times of 2.5 days. The 7 L dewar could not be refilled while in the shield, and once the geometry was disrupted new calibrations were required. The low capacity of the 7 L dewar lead to the purchase of an Ortec 30 L dewar with cold finger. This can be refilled online (while counting) and has a LN2 capacity that lasts approximately 10 days (see Figure 3.3).

3.3.5 Calibration Sources

Gamma-ray spectrometers must be regularly calibrated. Energy calibration can be done with any known source and its well documented emission spectrum. The activity of the source need not be known. Efficiency calibration is more complicated. Efficiency takes into account the source-to-detector geometry, so a particular calibration is only valid for its geometry. Calibration of efficiency also requires precise information about a source’s activity at a specific time. This information can be used with the exponential decay law to determine the source activity during the calibration. Calibration sources are commercially available. They typically include certified

41 Figure 3.4: Calibration source used for spectrometer energy and eﬃciency calibrations. measurements of their activities and their uncertainties. A set of eight sources (22Na, 54Mn, 57Co, 60Co, 109Cd, 133Ba, 137Cs, and a multinuclide) purchased from Eckert & Ziegler Isotope Products Laboratories were used for this work. The sources were ∼1 µCi with single and multiple peaks that spanned a range of typical energies (88–1836 keV).

3.3.6 Sample Holder

Since eﬃciency is dependent on source-to-detector geometry, it is important to maintain the same geometry during calibration as during measurement. This can be done with a sample holder. Presumably sample holders are commercially available, although none were found. A sample holder was designed and fabricated to meet the needs of this foil irradiation work. The sample holder consists of a base, two slotted walls, sample trays, and support rods (see Figure 3.5). The base mounts to the detector end cap with thumb-tightened nylon screws. The slotted walls allow sample trays to be loaded at various heights above the top of the detector. Two of the trays were designed to be used in conjunction. The calibration tray locates the calibration sources centered at the same positions as the foil tray. Due to the radial symmetry of some of the 21 foil locations, only ﬁve calibration locations are required. The trays have ridges and grooves that colocate the vertical axis midpoint of the calibration source with a 1 mm thick foil. A third tray that includes a radial locating

42 Figure 3.5: Exploded view of plastic sample holder designed and fabricated for this work. pattern was designed for the possibility of irregularly shaped samples. The entire sample holder is constructed out of clear plastic because high-Z materials attenuate gamma-rays, and metals often include radioactive trace elements.

43 Chapter 4

Computational Tools

This chapter explains the computational tools used to develop the high-energy ﬂux spectra measurement method at the Spallation Neutron Source (SNS). Suitable nuclear data libraries are also discussed. In addition to the GammaVision-32 software used for data acquisition, two major software packages and several custom programs were used for routine calculations.

4.1 Nuclear Data

4.1.1 ENDF

The standard nuclear data used in the United States come from the Evaluated Nuclear Data File (ENDF), the most recent version of which is the ENDF/B-VII.0 version [44]. Published in December 2006, ENDF/B-VII.0 contains data primarily for reactions with incident neutrons, protons, and photons on almost 400 isotopes, based on experimental data and theoretical predictions. The standard ENDF library extends up to 20 MeV for incident neutron data. The ENDF/B-VI.6 library released in 1996, included extensions up to 150 MeV maximum energy for about 40 neutron and 40 proton reactions on certain target isotopes, supporting spallation neutron source design and other applications. These extensions were the LA150 library. The higher energy cross section data were based upon measurements and on GNASH nuclear model code predictions. Unfortunately, a bug was later found in

44 Figure 4.1: Reaction cross section data for 27Al. Note the discontinuities at 20 MeV for the ENDF/B-VII.0 data, which are not present in the TENDL-2010 data. the GNASH code that was used to predict secondary emitted particle production. For ENDF/B-VII.0 the high energy LA150 data was recalculated and updated with newly corrected results. Even with the extended energies, ENDF libraries present several issues that make high-energy activation calculations difficult. The byzantine structure of ENDF files (known as ENDF-6 format) are a legacy from an era when data storage was costlier and scarcer than it is today. Consequently, neutron induced reaction cross sections are stored in multiple formats that must be recombined before they are used. For energies less than 20 MeV these reaction cross sections are stored in one format (called “File 3” in ENDF terms), while above 20 MeV they are stored in a different format (“File 5”). The different formats not only make their simultaneous use cumbersome, the two formats actually represent different physical values. Below 20 MeV, reaction cross sections can be extracted for a specific channel on a specific target, for example 27Al(n, α). Above 20 MeV, reaction cross sections can only

45 be extracted for a specific residual (product), for example 27Al(n, X)24Na. The two are not equivalent. The 27Al(n, X)24Na cross section includes the (n, n3He) reaction, whereas the 27Al(n, α) cross section does not. Also, the 27Al(n, α) cross section includes the 24mNa residual, whereas the 27Al(n, X)24Na cross section does not (Figure 4.1). Because the reaction cross sections above and below 20 MeV are different, it is not surprising to find a discontinuity in the reaction cross sections at that energy. This is the case for all reaction cross sections that span the artificial 20 MeV boundary, although it is more evident for some cross sections. Above 20 MeV the cross sections are usually calculated, whereas below 20 MeV cross section data primarily comes from evaluating experimental data. No attempt was made to match the ENDF data across the 20 MeV boundary. To quote one ENDF expert regarding the 20 MeV discontinuities, “They are unphysical, but they are OK for particle transport calculations, and that was their original intention [3].” Clearly, the ENDF libraries are less than ideal for high-energy neutron activation calculations.

4.1.2 LA150

As mentioned in Section 4.1.1, the LA150 cross section data have been recalculated and incorporated into the most recent release of the ENDF library. Separate LA150 cross section data are also available for use with MCNPX, so they are briefly mentioned here. The LA150 library was developed at Los Alamos National Laboratory for proton and neutron reactions up to 150 MeV on high priority nuclides for accelerator driven systems. The upper limit of 150 MeV was chosen partially because it corresponds to the pion production threshold, which is not predicted by the GNASH code. The GNASH nuclear model reaction code was used extensively in the LA150 library to predict nuclear reaction cross sections because measured data at higher energies are relatively sparse. Prior to the LA150 library, the intranuclear cascade model in the LAHET code was used to predict nuclear interactions above 20 MeV. LAHET is most accurate above 150 MeV. In that range, nuclear interactions are less sensitive to specific details of nuclear structure along with quantum effects in the scattering [45]. The LA150 library is in the ENDF-6 format, so it suffers from the same defi-

46 ciencies that aﬀect the ENDF library. Because the ENDF/B-VII.0 has been corrected and updated, the LA150 library is less useful than the ENDF library, but neither library is ideal for calculating high-energy neutron activation.

4.1.3 LAHET in MCNPX

The LAHET nuclear model code is available for use with the MCNPX software. Nuclear models are used by default in MCNPX when cross section tables are unavailable. Unfor- tunately, FM and DE-DF cards1 cannot be used with tallies using the LAHET model. As far as can be determined, this prohibits direct activation calculations in MCNPX using the LAHET code.

4.1.4 TENDL-2010

The TENDL-2010 cross section library is based on calculations by the TALYS 1.2 nuclear reaction simulation code, which uses evaluated data as well as nuclear models. TENDL- 2010 cross sections range up to 200 MeV with no artiﬁcial boundary or discontinuities at 20 MeV. In the incident neutron sub-library, data are available for the entire range of energies by reaction residual (Figure 4.1). For activation calculations, the most useful data is cross section by residual because ultimately the reaction producing the residual (i.e. (n,α) or (n,n3He)) is inconsequential to the foil activity measurement. The TENDL cross section data have several other features. TALYS generates libraries that are consistent, complete, and reproducible. TENDL libraries include covariance data. The data are freely available for download in plain text, ENDF-6 format, ACE format (used by MCNPX), and AMPX format. The TALYS-2009 data library was validated with a total of 686 International Criticality Safety Benchmark Evaluation Project (ICSBEP) benchmarks and was found to yield results closer to the experimental values than other libraries (i.e. JEFF-3.1 and Japanese Evaluated Nuclear Data Library (JENDL)- 3.3) [46]. The validity of TENDL at high energies is less veriﬁable due to the dearth of experimental data in that range.

1Commands in MCNPX are referred to as “cards.” See Section 4.2.1 for details.

47 4.1.5 Other Cross Section Data

Other sources of cross section data are briefly mentioned here. These sources were considered, but not used in this work. The European Activation File (EAF) is an ongoing project to produce data for the European fusion program. Its primary use is activation rather than transport. The EAF-2010 library contains 66,256 excitation functions involving 816 different targets from atomic numbers 1–100 in the energy range 10−5 eV–60 MeV [47]. While EAF-2010 is not available outside of fusion research projects, a similar library, EASY-2005.1, is available from the Radiation Safety Information Computational Center (RSICC) [48]. Although EAF is appropriate for activation calculations, 60 MeV is too low to use for SNS high- energy neutron spectrometry. The Japanese Evaluated Nuclear Data Library (JENDL) high energy file JENDL/HE- 2007 was developed to meet the intermediate and high energy nuclear data needs in accelerator-driven transmutation systems, particle radiation therapy, radioisotope production, space development, and other areas. The library extends up to to 3 GeV on 132 priority nuclides. Benchmark tests have been performed to validate the JENDL/HE-2007 since its release, such as deep-penetration of high-energy neutrons in shielding materials, and spallation experiments with high-energy protons. The JENDL/HE-2007 evaluation generally showed good agreement when compared with the other evaluations and experimental data [49]. Due to the high upper energy limit of the JENDL/HE-2007 library, its use in high-energy neutron spectrometry at SNS is appealing. Unfortunately, JENDL/HE-2007 has fewer activation cross sections than the TENDL library, and it suffers from the same 20 MeV artificial boundary issues as the ENDF library.

4.1.6 HILO2k

HILO2k is a coupled neutron-photon high-energy multigroup cross section library. The library was developed for SNS shielding design and other associated radiation transport problems. The library includes data for 32 nuclides, in 83 neutron groups and 22 photon groups ranging up to 2 GeV (Table 4.1). Neutron cross sections above 20 MeV are based on high-energy doubly diﬀerential neutron interaction and production data generated with

48 Group Upper Energy Group Upper Energy Group Upper Energy # MeV # MeV # MeV 1 2000 23 350.00 45 14.9200 2 1850 24 325.00 46 14.2000 3 1700 25 300.00 47 12.2100 4 1550 26 275.00 48 10.0000 5 1400 27 250.00 49 8.6070 6 1300 28 225.00 50 7.4080 7 1200 29 200.00 51 6.0650 8 1100 30 180.00 52 4.9660 9 1000 31 160.00 53 3.6790 10 950 32 150.00 54 3.0120 11 900 33 140.00 55 2.4660 12 850 34 120.00 56 2.3460 13 800 35 100.00 57 2.2320 14 750 36 80.00 58 1.8270 15 700 37 70.00 59 1.4230 16 650 38 60.00 60 1.1080 17 600 39 50.00 61 0.8209 18 550 40 40.00 62 0.7427 19 500 41 30.00 63 0.6081 20 450 42 25.00 64 0.4979 21 400 43 19.64 65 0.3688 22 375 44 16.91 66 0.2472

Table 4.1: HILO2k neutron energy groups from 247.2 keV to 2 GeV. a modified version of MCNPX, while below 2 MeV cross sections are based on ENDF data [50]. The HILO2k cross section libraries cannot be used for the high-energy neutron activation calculations needed for the foil measurements. The significance of HILO2k is its energy group structure, which is the standard group structure for neutron transport calculations at the SNS. The source terms used in MCNPX to calculate beamline fluxes at SNS are expressed using this group structure. As shown in equation 2.48, the flux spectrum that is unfolded from the foil measurements consists of group fluxes over several energy groups. The flux spectrum can be expressed using any energy group structure, but is best to use a group structure well suited to the expected features of the spectrum. Since the HILO2k group structure is already in use at the SNS, this is the group structure that is adopted for this neutron spectrometry work.

49 4.2 Software

4.2.1 Monte Carlo N-Particle eXtended (MCNPX)

MCNPX is a general purpose Monte Carlo radiation transport code designed to track many particle types over broad ranges of energies [51]. The software is very flexible and capable of modeling many types of radiation transport problems. In this work, MCNPX was used to calculate the response matrix elements, estimate expected reaction rates, and calculate gamma-ray self-shielding of the foils. One of several computer clusters at ORNL were used to process the MCNPX models. Two primary components are required to perform calculations in MCNPX: a source, and a tally on a material cell. The source can be parameterized by several variables including spatial location, energy, and direction. The material cell can also have various attributes including shape, location, and material composition. MCNPX offers several types of tallies, including the so-called F4 tally which specified as the “track length estimate of cell flux” [52]. The F4 tally is a Monte Carlo estimation of the average flux in a volume ˆ φV : 1 Z Z Z Z φˆ = dE dt dV dΩ Ψ(~r, Ωˆ, E, t) (4.1) V V where Ψ is the angular flux as a function of position ~r, direction Ω,ˆ energy E, and time t. Each particle that enter the cell scores:

T Score = W l (4.2) V

W is the particle weight, Tl is the particle track length in the cell, and V is the cell volume. Although the F4 tally is typically referred to as a “ﬂux” tally, its actual representation depends on the units of the speciﬁed neutron source. If the neutron source is in units of particles particles second , the F4 tally is in units of cm2second [52]. MCNPX has only a few tally types, but other quantities can be directly computed by using the FM or “Tally Multiplier” card. The FM card is used to calculate any quantity of the form: Z C φ(E)Rm(E)dE (4.3)

50 where φ(E) is the energy dependent ﬂux , Rm(E) is an MCNPX speciﬁed energy dependent response function, and C is a constant multiplier. For F4 tallies only, if C is negative, C atoms is replaced by |C| times the atom density (units of barn−cm ) of the cell where the tally is made. The response function can be a microscopic cross-section σm of a particular reaction cross cross-section, for example (n,2n). When the FM card is used with an F4 type tally, the reaction-rate can be computed.

Z Reaction Rate = Nρ × V φ(E)σm(E)dE (4.4)

where Nρ is the atomic number density. Another feature of MCNPX is the DE-DF “Dose Energy - Dose Function” cards. This feature allows the user to input an energy dependent point-wise response function to modify a tally. The DE-DF cards by default are logarithmically interpolated between points, but linear interpolation can be speciﬁed. Two features of MCNPX allow the user to bin the tally by energy. The “Tally Energy” En card allows tallies to be binned by groups of energies of the scoring particles. The “Special Treatment” cards FTn SCX allows tallies to be binned by particle source energy distribution. The distinction between these two cards is subtle but important. The En card bins the tally by particle energies where they “score”, but the FTn SCX card bins the tally by particle energies how they were “born.”

4.2.2 MAXED

Spectral unfolding is done with UMG 3.3. This software includes three main components: the MAXED unfolding program, the GRAVEL unfolding program, and the Integral Quan- tities Utility (IQU) auxiliary program. The programs were initially used with the Windows operating system and later compiled from Fortran source code for the Mac OSX operating system. MAXED is the primary unfolding program used for this work. MAXED applies the maximum entropy principle to unfold neutron spectrometric measurements. It was originally written for the unfolding of Bonner sphere data, but later modiﬁed to accommo- date proton recoil measurements. Advantages of MAXED include:

51 • it is the only mathematically consistent method of incorporating a priori information into the solution;

• it makes use of the estimated variances for each detector’s count rate in the unfolding process;

• the algorithm leads to a solution spectrum that is always a non-negative function

• the solution spectrum can be written in closed form [53].

Disadvantages of MAXED, based on the author’s personal experience and anecdotal con- versations [54], are that the documentation is sparse and the user interface is poor. When a neutron spectrometry measurement is made, some information is known about both the neutron source and about the physics that govern the neutrons interactions. This a priori information, usually derived from calculations or independent measurements, needs to be incorporated into the deconvolution process in a mathematically consistent way. In the language of MAXED, the initial estimate of the spectrum that incorporates all the a priori information is called the “default spectrum.” The challenge of neutron spectrum unfolding is to take the default spectrum and to modify it as minimally as possible so that it becomes consistent with the measurement data [55]. This minimally modiﬁed spectrum that is consistent with the measurements is called the “solution spectrum.”

Principle of Maximum Entropy

Derivation of the principle of maximum entropy is beyond the scope of this work, but it is based in part on the “cross entropy” function. Given a probability distribution p(x) and a distribution q(x) the cross entropy SCE is deﬁned as:

Z q(x) S = − q(x) ln dx (4.5) CE p(x)

The cross entropy SCE ≤ 0 and SCE = 0 iﬀ q(x) = p(x). Therefore, the cross entropy has been regarded as a measure of the amount of information required to change p(x) into

q(x). Applied to unfolding, the spectrum which maximizes SCE (or minimizes |SCE| since def SCE ≤ 0) for the default spectrum p(x) = f is the minimally modiﬁed solution spectrum q(x) = f.

52 The set of possible solution spectra are deﬁned by two constraints. First, the sum of the solution spectrum fn elements multiplied by the response matrix elements Rmn yields the measurement Mm within some discrepancy m:

N X Mm + m = Rmnfn (4.6) n

This is similar to equation 2.48, where the response matrix was deﬁned. The second constraint is that the discrepancies m are bound by some value Ω:

M X 2 m = Ω (4.7) σ2 m m

The value of Ω is chosen by the user in MAXED. In equation 4.7, the discrepancies are squared to be positive and normalized to the standard uncertainty of each measurement σm. Setting Ω to the number of detectors M essentially requires that the average uncertainty- normalized discrepancy is equal to 1:

M 1 X 2 m = 1 (4.8) M σ2 m m

Equations 4.8 and 4.6 state that the discordance between the calculated ﬂux spectrum and the measurements should be the same as the measurement uncertainty on average. It’s reasonable that the solution spectrum cannot be more precise than the measurements that were used to determine it. The solution spectrum satisﬁes equations 4.7 and 4.8, and maximizes the entropy described by equation 4.5:

N X fn Def S = − fn ln Def + fn − fn (4.9) n fn

Def Equation 4.9 is a more general form of 4.5 with two additive terms. The addition of fn ensures that the global maximum of S is zero, which is convenient but not necessary. The subtraction of fn ensures that in the absence of any other constraints, when S is maximized, Def fn = fn rather than just being proportional [55].

53 Equation 4.9 can be maximized using the calculus of variations and Lagrange multipliers λm. Using this technique, solving for the the solution spectrum and the discrepancies yields:

M ! DEF X fn = fn exp λmRmn (4.10) m 1/2 2 ! λmσm 4Ω m = (4.11) 2 PM 2 m (λmσm)

Equations 4.10 and 4.11 show that the solution spectrum and the discrepancies can be expressed in terms of the default spectrum, known parameters, and Lagrange multipliers in closed form. This is diﬀerent from other iterative techniques where a solution is expressed in terms of the previous iteration. Equations 4.10 and 4.11 allow for straightforward propagation of errors and sensitivity analysis that is not possible with other iterative solutions. The maximization of the entropy in equation 4.9 is equivalent to maximizing a potential function Z:

N M ! M !1/2 M X Def X X 2 X Z = − fn exp − λmRmn − Ω (λmσm) − Mmλm (4.12) n m m m

Def In equation 4.12, the fn , Rmn, Mm, σm and Ω are speciﬁed as input values. For any chosen λm values, Z can be evaluated. Therefore, to ﬁnd the global maximum Z, the space of all λm must be sampled methodically. This is accomplished using the Simulated Annealing (SA) algorithm.

Simulated Annealing (SA)

Simulated annealing is analogous to the physical cooling of materials. When some materials are allowed to cool slowly, they form large crystals billions of times larger than the constituent atoms. The crystal is global minimum of energy for the system, and it forms because the atoms are allowed to redistribute themselves as they lose energy. If the material is cooled rapidly, a local minimum is reached, but not necessarily a global minimum. Nicholas Metropolis and his colleagues incorporated the physical principles of annealing

54 into numerical calculation. Metropolis algorithms such asSA allow for redistribution out of local extrema by probabilistically selecting best estimates using an artificial temperature function [56]. i i i As used in MAXED, theSA algorithm randomly chooses a trial point ( λ1, λ2, ..., λN ) within some step length of the starting point. If the potential function in equation 4.12 evaluates to a larger value, the “uphill” trial point is always accepted as the new best estimate. If the potential function evaluates to a lower value, the “downhill” trial point is accepted with a probability p(T ) as a function of artificial temperature. The lower the temperature, the less likely downhill points will be accepted. The process iterates, and a new i+1 i+1 i+1 random trial point (λ1 , λ2 , ..., λN ) some step length from the current best estimate is again evaluated. After a number of random trials at a given temperature, the artificial temperature will decrease according to a cooling schedule. At the lower temperature, the step length decreases as does the probability p(T ).SA is able to move out of local maxima, but as it progresses, it samples from the most promising region of the space. TheSA algorithm used by MAXED is a modification of the SIMANN.F program [57]. Many of the parameters of theSA algorithm were optimized by the MAXED authors. The user sets the initial temperature and a temperature reduction factor. The optimal initial temperature depends on the particular problem. The MAXED authors recommend selecting an initial temperature high enough such that no new maxima are found in the first set of iterations at constant temperature, noting that 1000 is typically large enough [55]. In practice, this has not always been achievable. Some unfoldings with MAXED have shown 1 new maximum in the first iterations at constant temperature even for temperatures as high as 1025. The reason this occurs is not known, but it could be due to a bug in the program. In these cases, the initial temperature is selected when it is high enough such that < 2 new maxima are found initially. In each step of the cooling schedule, the temperature reduction

factor RF is applied such that Tnew = RF × Told. A temperature reduction factor of 0.85 is recommended by the MAXED documentation and was used for all unfoldings.

55 4.2.3 GRAVEL

The GRAVEL algorithm was not used extensively for this work, but is explained here briefly to contrast against MAXED. GRAVEL is an extension of the SAND-II algorithm [58]. SAND-II assumes an estimate of the flux, and for subsequent iterations applies a nonlinear j adjustment [8]. Given an element of a discretized flux spectrum fn for iteration j, the flux for iteration j + 1 is defined as:

PN j Mm  Wmn ln n PM R f j f j+1 = f j exp m mn n (4.13) n n  PN j  n Wmn R f j M2 W j = mn n m (4.14) mn PN j σ2 n Rmnfn m

The default spectrum is used for the ﬁrst iteration. Given a non-negative default spectrum, the solution spectrum will always be non-negative. Because equation 4.13 is not written in closed form similar to equation 4.10, the same uncertainty propagation and sensitivity analysis that is used with MAXED cannot be used with GRAVEL [59].

4.2.4 Integral Quantities Utility (IQU)

The IQU program calculates the uncertainty in the solution spectrum ∆f Meas due to the uncertainty in the measurements, the uncertainty in the solution spectrum ∆f Def due to the uncertainty in the default spectrum, and the total uncertainty ∆f Total in the solution spectrum due to both uncertainties. The N ×N uncertainty matrix UMeas for the solution spectrum due to the uncertainty in the measurements is:

δf δf T UMeas = KMeas (4.15) δM δM

Meas δf where K is the M × M measurement covariance matrix, δM is the N × M sensitivity δf T matrix, and δM is the M × N transpose of the sensitivity matrix. Similarly, the N × N uncertainty matrix UDef for the solution spectrum due to the uncertainty in the default spectrum is: δf δf T UDef = KDef (4.16) δf Def δf Def

56 Def δf where K is the N ×N default spectrum covariance matrix, δf Def is the N ×N sensitivity δf T matrix, and δf Def is the N ×N transpose of the sensitivity matrix. The total uncertainty UTotal is just the sum of the two uncertainty matrices:

δf δf T δf δf T UTotal = KMeas + KDef (4.17) δM δM δf Def δf Def

The group uncertainties are found by multiplying the uncertainty matrices by the solution spectrum:

∆f Meas = UMeasf (4.18)

∆f Def = UDef f (4.19)

∆f Total = UTotalf (4.20)

Because the measurements are independent, the measurement covariance matrix δf δM is assumed to be diagonal with σm on the diagonal. For the default spectrum covariance matrix KDef , a diagonal covariance matrix is also assumed. Although a block diagonal matrix may be appear to be justiﬁed since the variation in the ﬂux of a particular energy group may be correlated with its adjacent groups, the Unfolding with Maxed and Gravel (UMG) documentation notes:

...this approximation is adequate for the case of maximum entropy deconvolution, as can be seen by looking at the mathematical form of the maximum entropy solution. Particular cases using data from measurements show that the oﬀ-diagonal elements are 2 or 3 orders of magnitude smaller than the diagonal elements [58].

The “uncertainty” in the default spectrum is input by the user, and is not a true measurement uncertainty, rather it is a judgement of the default spectrum’s accuracy. The sensitivity matrices are too complicated to present here, but they can be expressed in terms of the solution spectrum, response matrix, and other input parameters in closed form. The sensitivity and uncertainty matrices are computed by the IQU program from the input parameters and the solution spectrum. The IQU program can also assign uncertainty to any “integral quantity” H of

57 the form: N X H = hnfn (4.21) n

For example, the integral quantity H could be the dose, and hn could be a group-wise ﬂux-to-dose conversion factor. The uncertainty in the integral quantity ∆H is given by:

sX ∆H = hiUijhj (4.22) i,j

4.2.5 Other Unfolding Methods

Several methods of deconvolution have been applied to neutron flux spectrum unfolding. These methods are briefly noted here. A comparison of different unfolding techniques found that some algorithms are perform better in different energy regimes, perform better with different initial spectra, can combine different methods of detection, and are computation- ally faster [60]. Neural Networks are algorithms for cognitive tasks, such as learning and optimization, which are loosely based on concepts derived from brain research. They simulate a highly interconnected, parallel computational structure with many individual processing elements, or neurons. These networks learn through an iterative process of adjustments using training sets. After a network has been trained, it operates on input data to produce new output. Neural Networks have been used to unfold Bonner sphere data [61, 62], and foil activation data [34]. Genetic algorithms mimic natural evolution by “breeding” candidate solutions. Random breeding occurs over many generations with a bias toward “high-fitness” candidate solutions. When solutions breed, they exchange components to produce offspring solutions, and then they are “killed off” to maintain a constant population size. After many generations of breeding, the fittest solutions are expected to have survived. Genetic algorithms have been used to unfold neutron spectra from Bonner sphere data [63,64]. The BUNKI code is based on the BON algorithm, which applies the transpose of the response matrix to both sides of equation 2.48 to determine a physically relevant solution iteratively. The solution is not sensitive to initial guess, so it is best used when a priori information in insufficient [65]. Due to this insensitivity, the BUNKI code was

58 favored when compared to the MAXED code in unfolding Bonner spheres [66]. This algorithm is similar to the SAND-II algorithm and the results obtained by the two codes have been found to be nearly identical [67]. Gold’s algorithm, also known as the ratio method, requires no a priori information. The measured data and the response matrix can be used to create an initial approximation, which is further iteratively refined. The algorithm is implemented in the SAYD code, which was found to agree well with other codes in a computational benchmark [68]. The Monte Carlo method can also be applied to neutron spectrum unfolding as well as neutron transport. A large number of spectra are generated randomly and the response fitting errors are calculated. The mean value of the best few solutions is used to smooth statistical fluctuations [69]. This method is regarded as time consuming, but can be used for parameter estimation in cases where other techniques fail [70]. The code MINUIT has used a multi-parameter function minimization routine for the unfolding of activation foil data. A fast breeder reactor neutron spectrum was unfolded using the MINUIT code and a set of experimental activation rates without a guess solution. The solution was found to be in good agreement with measured reaction rates. The output spectrum was less dependent on the cross-section variations of the activation foil detectors, a unique advantage [71].

4.2.6 Custom Programs and Additional Software

Several custom programs were written by the author to streamline the data analysis process. A brief description of each program follows.

• get tally files.pl - A Perl script written to parse the MCNPX output ﬁle for the relevant information and then output the information to freely formatted text ﬁles usable by UMG MakeFiles.

• UMG MakeFiles - A Fortran-90 compiled executable that writes three files used as input to the MAXED program. MAXED requires strictly formatted Fortran-style input text files. Experience has shown that formatting these files “by-hand” is ex- tremely tedious. UMG MakeFiles uses the output files of the get tally files.pl

59 script and a text ﬁle containing the default spectrum (one energy group to a line, whitespace delimited) to produce the .IBU, .FLU, and .FMT ﬁles.

• PowerIntegrator.py - A python program written to read a formatted text ﬁle containing beam power data and a ﬁle containing decay constants, and then integrate over a time range to compute the beam correction factors (Section 4.3.4).

• FLU Metrics.py - A python program that compares two .FLU (MAXED flux spectrum) files and calculates four metrics based upon their differences. The four metrics

are the `1 or “taxi-cab” norm, the Euclidean norm, the average relative difference, and an absolute difference dose norm. The absolute difference dose norm uses flux- to-dose conversion factors contained in a text file (Section 5.1.3).

• diffPlotter.py - A python program that produces three plots comparing two .FLU files. The first plot is both flux spectra on the same scale, the second plot is the absolute difference between the two, and the third plot is the relative difference between the two.

• GVAssembler.py - A python program that takes multiple .SPE (GammaVision-32 ASCII formatted spectrum) files and assembles them into a single file. The program sums the counts in each channel and accounts for the difference between start and stop times by adding the difference to the total dead time.

• XS Plotter.py - A python program that takes an ASCII formatted cross section file and plots the data. The program also reads an energy group file and finds the group-averaged cross section for each energy group.

• SiSp plotter.py - A python program that plots Si-Sp (“Source Information - Source Probability”) cards from MCNPX. This is useful to visualize MCNPX input such as source energy distributions.

Additional software has been used to do some analysis. Prior to using SciPy, OriginLab was used for some plotting. Matlab had been used to do some initial investi- gations of the gamma-ray spectrum. Microsoft Excel spreadsheets are used for simpliﬁed

60 quick estimates and some data storage. ImageJ was used to do image analysis of autora- diographs.

4.3 Application of Software to Speciﬁc Problems

4.3.1 Estimating of Foil Activity Using MCNPX and Other Tools

Calculating expected results before executing an experiment is prudent. Foil activity calculations help to estimate type and number of foils, the irradiation time, the counting time, and other important parameters. To estimate foil activity, two methods were used: a cursory calculation using a spreadsheet, and a detailed calculation using MCNPX. The first step of the cursory calculation is to get average group cross-sections from the continuous cross sections. Continuous reaction cross sections are available for download in plain text from the TENDL website. Using the HILO2k group structure and the custom XS Plotter.py program, group averaged cross sections σn are computed quickly. Multigroup source terms that SNS staff have calculated are available [2]. To find the flux at a beamline sample location φn, MCNPX can be used with a flux tally, or a simple 1/r2 factor can be applied, where r is the source to sample distance. It may also be necessary to scale the source term by a beam power factor. The multigroup fluxes and the multigroup cross sections can be multiplied group by group along with the number of target nuclei Ni in the foil to produce a reaction rate

G X Reaction Rate = Ni φgσg (4.23) g

This can be used to calculate the activity at the end of the irradiation, which depends on the irradiation time t0 and the decay constant λ:

G −λt0 X A(t0) = 1 − e Ni φgσg (4.24) g

The total number of counts C for a gamma-ray of energy Eγ is calculated during the count

61 time, which occurs between tstart and tstop:

−λt −λt G e start − e stop X C(t , t ) = I (E ) 1 − e−λt0 N φ σ (4.25) start stop γ γ λ i g g g where Iγ is the gamma emission probability, and (Eγ) is the spectrometer eﬃciency. Equation 4.25 is similar to equation 2.19, but equation 4.25 does not account for gamma- ray self-shielding. The following equation is used to estimate the Relative Standard Error (RSE) of the count: r 1 RSE= (4.26) C

Equations 4.26 and 4.25 can be used to make decisions about the number of foils, the irradiation time, the counting time, and other parameters. The detailed calculation uses MCNPX. Foil geometry and orientation can be modeled at the sample location and the source can be modeled using the SNS source terms. The reaction rates can be calculated by using an F 4 tally on each foil cell along with an FM multiplier card, where the reaction cross section is the response function and the constant multiplier accounts for the number of foil nuclei and units conversions. The MCNPX calculation uses the continuous energy cross sections and accounts for changes in the ﬂux in the foils (self-shielding). A more accurate result is obtained by calculating equation 4.4 instead of equation 4.23. This result can be substituted into equations 4.24 through 4.26 to obtain more accurate count estimates. The cursory method works well for scoping calculations and quick estimates, while the detailed calculation should be used when modeling actual experimental data.

4.3.2 Calculation of Response Matrix Elements Using MCNPX

MAXED solves the inverse problem of equation 4.6, which is to say that given the Mm and the Rmn, the fn are determined. As stated in Section 2.4, it is convenient to express the Mm as the reaction rate during irradiation:

Z Emax Mm = Niσ(E)φ(E)dE (4.27) Emin

62 Combining (4.27) with the deﬁnition of Mm in equation (2.44), the energy dependent response function R(E) is determined:

R(E) = Niσ(E) (4.28)

Using the definition of Rmn in equation (2.45), the response matrix elements are determined by: R En+1 N σ(E)φ(E)dE En i Rmn = (4.29) R En+1 φ(E)dE En This is just the group reaction rate divided by the group flux, which conforms to the original definition of the response function R = M/φ from equation 2.43. As explained in Section 4.3.1, the reaction rate for each foil m can be calculated in MCNPX using F 4 tallies and FM tally multiplier cards. These results can be binned by energy group n using the FTn SCX special treatment card. The response matrix elements are calculated when these m × n MCNPX results are each divided by the appropriate group flux, which can be done post-processing. This is final step is done with the custom program get tally files.pl. Calculation of the response matrix elements from equation 4.29 appears straight-

forward, but unfortunately the Rmn are defined in terms of the unknown flux φ(E). An estimate of the flux must be used, and an appropriate estimate is the default spectrum to be used by MAXED. Once the new solution spectrum is found, that solution spectrum can be used to recompute the response matrix, and further unfoldings can be performed iteratively to refine the solution [67].

Sensitivity of the Response Matrix to Flux

In practice, the response matrix is not very sensitive to differences in the flux used to calculate it. The N energy groups should be defined such that the flux is approximately constant across each group, so equation 4.29 can be approximated by:

R En+1 N σ(E)dE En i Rmn ≈ (4.30) En+1 − En

63 Equation (4.30) can be rewritten in more intuitive forms:

R En+1 σ(E)dE En Rmn ≈ Nj En+1 − En n Rmn ≈ Nj σ (4.31)

n Rmn ≈ ΣV (4.32) where nσ is the average microscopic reaction cross section for group n, nΣ is the average macroscopic reaction cross section for group n, and V is the volume of the foil. Equation

4.31 is commensurate with equation 4.28 and is a good approximation of Rmn. But even if the ﬂux were exactly constant across each group, equation 4.31 would fail to be exact because the eﬀect of the foil thickness has not yet been considered.

Accounting for Neutron Self-Shielding

Ideally, a foil material and thickness is chosen such that it does not significantly disturb the flux. A single thin foil approximates the ideal situation, but sometimes a foil is not thin or multiple foils are layered into a thick foil packet. Due to its finite thickness and cross section, the front of a foil will slightly shield the rear of a foil from neutrons. To account

for this neutron self-shielding, a distinction is made between the ﬂux in the foil φF oil(E) and the ﬂux at the sample location in the absence of the foil φ(E).

R En+1 N σ (E)φ (E)dE Foil En j j F oil Rmn = (4.33) R En+1 φ(E)dE En

MCNPX computes φF oil(E) when foil materials are speciﬁed. When the materials are

“voided,” MCNPX computes φ(E). The ﬂux in the foil φF oil(E) depends on thickness and the ordering of the foils, while φ(E) does not. For a given beamline, it is suﬃcient to

compute φ(E) once, while φF oil(E) will need to be recomputed for any change in the foil conﬁguration. Using the approximation of constant ﬂux across energy groups, equation 4.33 becomes:

φ FoilR ≈ Foil N nσ (4.34) mn φ j

64 Equation 4.34 has an additional ﬂux-dependent factor, as compared to 4.31. This term can be considered the neutron self-shielding factor. In summary, equation 4.33 with the default spectrum is used to calculate the response matrix elements in MCNPX, but equation 4.31 provides a good approximation which can be used to check the MCNPX output for reasonableness.

4.3.3 Calculation of Gamma-ray Self-Shielding Using MCNPX

Just as neutrons are attenuated as they propagate through the foil, so are the gamma-rays which emanate from the foil. Generally, gamma-ray self-shielding is a greater effect in a foil than is neutron self-shielding. The effect of a millimeter of high-Z (atomic number) material can be significant, especially for softer gamma-rays. To calculate the gamma-ray self shielding effect, foils of various geometry and material are modeled in MCNPX. The gamma-ray source is specified to originate from the bulk of the foil using a single energy. An enveloping surface, such as a sphere, surrounding the foil is used to tally the particle current (an F 1 tally). By default, tallies are in units of per source particle. When this tally is binned by energy using the En card, the uncollided particle current is distinguished from the collided current. The gamma-ray self-shielding factor is the uncollided current tally. Several runs over a range of gamma-ray energies will produce several points which can be interpolated for an approximate gamma-ray self- shielding function Fγ(E).

4.3.4 Correcting for Time-Varying Flux

For a neutron flux that varies with time, such as when accelerator power changes from one minute to the next, the activation rate is not constant. Two important relationships simplify the analysis of this scenario. First, the magnitude of the flux changes with time, but the energy distribution does not. This is equivalent to saying the energy-time dependence of the flux φ(t, E) is separable:

φ(t, E) = F(t) φ(E) (4.35)

65 The F(t) is the time-dependent magnitude of the ﬂux φ(E). These variables can be separated because the underlying physical processes that generate the neutrons do not change when the beam power changes. The energy spectrum of neutrons is determined by the spallation and moderation processes. When the beam power increases or decreases, there are more or less of these processes occurring, but the processes themselves are unchanged, therefore the energy component of the spectrum is unchanged with beam power. The second relationship is that the time-dependent magnitude of the ﬂux F(t) is linearly proportional to the beam power P (t):

F(t) = k P (t) (4.36)

h 1 i where k is some yet undetermined scalar constant with units of power . Equations (4.35) and (4.36) can be combined into:

φ(t, E) = k P (t)φ(E) (4.37)

Using this time-dependent deﬁnition of the ﬂux, the change in the number of nuclei Ni due to the production, destruction, and decay processes is:

dN (t) i = production − destruction − decay dt dN (t) i = kP (t)N (t)σ φ − kP (t)N (t)σ φ − N (t)λ (4.38) dt j j i i i i

Where the simpliﬁed notation from equation (2.7) has been used. With initial condition

Ni(0) = 0 the diﬀerential equation (4.38) can be solved in a manner similar to the way equation (2.6) was solved. To make the derivation simpler, this time the two approximations will be applied before seeking the solution:

dNi(t) = kP (t) Nj(t) σjφ −kP (t)Ni(t)σiφ − Ni(t)λi dt | {z } | {z } ≈Nj ≈0 Z t0 −λit0 λit Ni(t0) ≈ Njσjφe kP (t)e dt 0

The foil activity Aφ(t)(t0) during irradiation at time t0 due to a time-dependent ﬂux φ(t, E)

66 is: Z t0 −λit0 λit Aφ(t)(t0) = Njσjφλie kP (t)e dt (4.39) 0 Note that when kP (t) = 1, equation (4.39) reduces to the time-independent equation (2.10).

Equation (4.39) can be rewritten in terms of the activity Aφ(t0) due to a time-independent ﬂux:

λ e−λit0 Z t0 −λit0 i λit Aφ(t)(t0) = Njσjφ(1 − e ) kP (t)e dt (1 − e−λit0 ) | {z } 0 Aφ(t0)

This equation can be rearranged so that the activity Aφ(t0) due to a time-independent ﬂux is written in terms of power history correction factor CP (P (t), t0):

Aφ(t)(t0) = CP (P (t), t0) Aφ(t0) (4.40)

−λit0 R t0 λit λie 0 kP (t)e dt CP (P (t), t0) = (4.41) 1 − e−λit0

Equation (4.41) permits activity analysis of a ﬂuctuating power foil irradiation by relating it to a constant power case. This correction requires the time dependence of the beam power P (t), which is known for a given irradiation. The integral in equation 4.40 is computed by the custom program PowerIntegrator.py.

4.4 Other Corrections

4.4.1 Partially Irradiated Foils

Autoradiography is the primary tool in determining if a foil has been partially irradiated. While a foil still retains signiﬁcant activity, it is exposed to an autoradiographic plate, generating two-dimensional image of its activity. This image is then analyzed using the ImageJ program. Tools in ImageJ allow pixel sizes to be calibrated to absolute length, so that total irradiation area can be determined. The ratio of the irradiated area to the total area is a fractional factor that can be applied to activity and reaction rate calculations used in the unfolding.

67 Figure 4.2: Dimensions of the sources used for calibration of the gamma-ray spectrometer [72].

4.4.2 Calibration Source Size Versus Foil Size

Although the sample holder attempts to colocate the axial and radial centers of the calibration source and foils using two trays, the calibration sources and the foils are not of the same dimensions. This introduces small errors into the efficiency calculation. The active volume of the calibration sources is a cylinder 5 mm in diameter and 3.18 mm high (Figure 4.2), while the foils are cylinders ∼10 mm in diameter and up to 10 mm high. The closer the sample is measured to the gamma-ray spectrometer, the more sensitive it is to differences in position. Sensitivity to changes in position can be empirically estimated from calibrations at various positions. Based on these calibrations, the differences in the calibration source size and foil size can safely be neglected.

68 Chapter 5

Measurement Results

This chapter details the experimental measurements and their results. Two foil irradiations were performed: at the ARCS beamline, and at the POWGEN beamline.

5.1 July 8, 2009 ARCS

5.1.1 Foil Irradiation

Foil Packet

The foil packet used at the ARCS beamline consisted of four types of foils: two aluminum, three copper, three nickel foils, and a single bismuth foil. These foils were selected because they were immediately available for the irradiation opportunity. The foils were mounted to an aluminum holder and wrapped in aluminum foil (∼0.033 mm thickness). The packet was mounted to a sample stick and placed in the beam at the ARCS sample location.

Beam Power

As discussed in Section 4.3.4 beam power on the target must be monitored to scale the calculated ﬂux spectrum appropriately. The power on target is measured using a proton beam current monitor in the ﬁnal beam transport section of the accelerator. Measure- ment uncertainty on this instrument is unavailable. Tens of thousands of control points, including the proton beam power, are recorded by the centralized controls system at the SNS. Once recorded, the values of these control points can be retrieved using software

69 Figure 5.1: Bismuth, copper, nickel, and aluminum foils along with aluminum holder as modeled in MCNPX and visualized with the VisEd (left) and Sabrina (right) software. from a system archive. Unfortunately, during the ARCS foil irradiation, the control system archive experienced some kind of system-wide error that prevented data recording. This error was confirmed with SNS control systems staff [73]. Therefore, no beam power data was archived during the irradiation for approximately 55 minutes from 12:10 until 13:15. The author does not recall significant power fluctuations during this time and none were noted in the log book, but beam power fluctuations may have occurred. Due to the control system archiver failure, the beam power history must be assumed during the 55 minute interval. The power history during this interval is assumed to be similar to the power history that was recorded by the control system archiver from 13:15 until 14:54. The power history from 13:15 to 14:54 was used to calculated the power history correction factor. The average power from 13:15 to 14:53 is 379.748 kW. The calculated correction terms are minor, in most cases less than 1% (Table 5.1).

70 Figure 5.2: Beam power during the July 8, 2009 ARCS foil irradiation.

Nuclide Correction Nuclide Correction

208Bi 1.000 200mBi 1.017 207Bi 1.000 199Bi 1.020 206Bi 1.000 199mBi 1.023 205Bi 1.000 198mBi 1.046 204Bi 1.001 57Ni 1.000 203Bi 1.000 58Co 1.000 202Bi 1.004 61Cu 1.002 201Bi 1.004 64Cu 1.000 201mBi 1.008 24Na 1.000 200Bi 1.014 27Mg 1.052

Table 5.1: Power history corrections for the ARCS foil irradiation. Corrections are deﬁned by equation 4.41.

71 Figure 5.3: Radiation control technician checks the dose rate on the foils as they are removed the sample position after irradiation.

Irradiation Time

The total irradiation time from 12:10 to 14:54 is 9840 ± 120 s.

Cooling Time

When the beamline shutter was closed at 14:54, irradiation ceased and the foil activity began to decay. A ﬁnite time interval was necessary to remove the foils and have a radiation control technician perform a radiological survey. The foils were then transported to the counting room, and several initial counts were performed, none of which were used for the ﬁnal analysis. The count used for the analysis began at 18:38:41, so the cooling time is approximately 3 hours and 44 minutes (13,380 ± 120 s).

72 Figure 5.4: Foil activity was measured on the detector end cap. A white index card was used to reduce possible contamination of the detector.

5.1.2 Gamma Spectrometry

Energy and Eﬃciency Calibrations

Energy and efficiency calibrations were performed using the following radioisotope calibration sources: 109Cd, 57Co, 123mTe, 51Cr, 113Sn, 85Sr, 137Cs. The energy calibration used 15 peaks ranging from 122 keV to 2754 keV. A linear fit was made to the data with a coefficient of determination r2 value of 0.999. The efficiency calibration used 5 peaks ranging from 159 keV to 835 keV. The data were used to produce a linear fit (natural log of efficiency vs. natural log of energy) with a coefficient of determination r2 value of 0.997. Because the samples were measured on the detector cap in a “close geometry,” True Coincidence Summing (TCS) was present. Only non-cascade gamma peaks were used for the calibration, which limited the number

73 Figure 5.5: Energy calibration curve for the HPGe spectrometer used for the ARCS foil irradiation. of peaks to 5. Efficiency data below 159 keV were not required for the measurements, but several efficiency measurements beyond 835 keV, where the curve is extrapolated, were made. When possible, the use of extrapolated curves is best avoided. Unfortunately, in this case no non-cascade gamma peaks above 835 keV were measured with the available calibration sources. Use of the extrapolated curve is justified because in the 130 keV to 2000 keV range, the efficiency curve (natural log of efficiency vs. natural log of energy) is expected to be approximately linear [10].

74 Figure 5.6: Eﬃciency calibration curve for the HPGe spectrometer used for the ARCS foil irradiation.

75 Count Time

The count used for the analysis was started at 18:38:41. Total Real Time: 148,745.00 ± 1 s (∼ 41.3 hours) Total Live Time: 147,618.00 ± 1 s (∼ 41.0 hours)

Candidate Peaks

Forty-one full-energy peaks from 10 nuclides were identiﬁed in the gamma-ray spectrum. These peaks will be referred to as “candidate peaks” because they are potentially valid peaks to use in the unfolding. Only a subset of the total peaks will be used in the unfolding. Some of the candidate peaks may have been misidentiﬁed or poorly measured. The time analysis is used to inform the selection of peaks to be used in the unfolding.

Nuclide Peak Energy Nuclide Peak Energy Nuclide Peak Energy keV keV keV 199Bi 841.70 204Bi 899.15 206Bi 620.48 204Bi 911.78 206Bi 632.26 201mBi 846.40 204Bi 918.15 206Bi 803.10 204Bi 984.02 206Bi 881.01 202Bi 422.18 206Bi 1018.63 202Bi 657.49 205Bi 703.44 206Bi 1098.26 202Bi 960.67 205Bi 1043.72 206Bi 1595.28 205Bi 1764.36 206Bi 1718.70 203Bi 820.30 205Bi 1861.67 203Bi 825.20 58Co 810.78 203Bi 1033.80 206Bi 183.98 203Bi 1536.40 206Bi 262.70 60Co 1173.24 203Bi 1670.50 206Bi 343.51 60Co 1332.50 203Bi 1847.40 206Bi 398.00 206Bi 497.06 24Na 1368.63 204Bi 374.72 206Bi 516.18 24Na 2754.03 204Bi 670.75 206Bi 537.45

Table 5.2: Candidate peaks in the gamma-ray spectrum of the ARCS foil irradiation.

76 Figure 5.7: The gamma-ray spectrum of the foils irradiated at the ARCS beamline.

77 Time Analysis

As a precaution, the gamma-ray spectrum was saved several times while counting. Later it was realized these data could be used to measure the decay constant of candidate peaks. Measuring the decay constant provided more evidence to determine if a peak was legiti- mately produced by the suspected nuclide in the foil. In other words, if the measured decay constant of a candidate peak is significantly different from the known decay constant, this peak can be eliminated as a candidate. Decay constants longer than a few days were difficult to measure with these low count rates and relatively short count times. In many cases, at least one peak per nuclide was measured with a relative error less than 10%.

78 (a)

(b)

Figure 5.8: Time analysis measuring the decay constant for the 202Bi and 203Bi candidate peaks.

79 (a) 204Bi

(b) 205Bi

Figure 5.9: Time analysis measuring the decay constant for the 204Bi and 205Bi candidate peaks.

80 (a)

(b)

Figure 5.10: Time analysis measuring the decay constant for the 206Bi candidate peaks.

81 (a) 58Co

(b) 24Na

Figure 5.11: Time analysis measuring the decay constant for the 58Co and 24Na candidate peaks.

82 Figure 5.12: Calculated ARCS group source spectrum.

5.1.3 Unfolding

ARCS Calculated Spectrum

A neutron source spectrum for the ARCS instrument was calculated and used for shielding design at the SNS[74]. This neutron source spectrum represents the best estimate for the high-energy beam at ARCS. The origin of the source is at the core vessel insert, approximately 1 m from the moderator face and 13.6 m from the sample position. The source area is rectangular (12 cm x 10 cm) with two angular distributions (0◦ < θ < 1◦ and 1◦ < θ < 2◦) each with an associated energy distribution. Only neutrons in the central angular distribution are transported uncollided to the sample location. The ﬂux at the sample location is computed using MCNPX and the calculated source spectrum.

83 Figure 5.13: Calculated ARCS ﬂux spectrum at the sample location.

Calculated Vs. Measured Counts

Using the calculated ARCS spectrum to compute reaction rates, then using the irradiation, cooling, and counting time along with the gamma-ray emission probability, spectrometer efficiency, gamma self-shielding, and power-history corrections, the expected number of counts were computed for each peak. If the calculated ARCS spectrum accurately reflects the measured spectrum, the ratio of the calculated counts to the measured counts should be approximately 1. In all cases, the average calculated-to-measured count ratio is greater than 1 and in most cases in the range of 3–5. This is an indication that the calculated spectrum does not accurately reflect the measured spectrum, but unfolding must be used determine a spectrum consistent with the measurements.

84 (a) 202Bi

(b) 203Bi

Figure 5.14: The ratio of calculated counts to measured counts for the 202Bi and 203Bi candidate peaks.

85 (a) 204Bi

(b) 205Bi

Figure 5.15: The ratio of calculated counts to measured counts for the 204Bi and 205Bi candidate peaks.

86 Figure 5.16: The ratio of calculated counts to measured counts for the 206Bi candidate peaks.

87 (a) 58Co

(b) 24Na

Figure 5.17: The ratio of calculated counts to measured counts for the 58Co and 24Na candidate peaks.

88 Figure 5.18: Flux-to-dose-rate conversion factor as a function of energy used to compute the dose rate metric.

Dose Rate Metric

To compare the similarity of one spectrum with a default spectrum, a dose rate metric was devised. The dose rate metric is the dose rate produced by the absolute ﬂux diﬀerences of each energy group divided by the total dose rate produced by the default spectrum.

P ˙ default ˙ |Hg − Hg| g=groups Dose Rate Metric = (5.1) H˙ default

The dose rate metric is essentially a normalized `1 norm (or “taxi-cab” norm) weighted by an energy dependent flux-to-dose-rate conversion factor. It is always greater than or equal to the difference in overall dose rates produced by the two flux spectra. The use of the absolute value of the difference of the group-by-group dose rate means that the dose rate metric will equal zero if and only if the two spectra are identical. The energy dependent flux-to-dose-rate conversion factor used is shown in Figure 5.18.

89 Initial Analysis

An initial spectral unfolding from 20 MeV to 120 MeV was done with the MAXED program using reaction rates determined from 5 measured nuclide activities: 206Bi, 205Bi, 204Bi, 203Bi, 202Bi. The initial default spectrum used for the unfolding was the calculated ARCS spectrum, and the response functions were determined from the group-wise reaction rates of the default spectrum. Initially, the calculated high-energy neutron flux at the sample position was used as the default spectrum in the unfolding, but later it was determined that using the source spectrum directly was more convenient. The results of the unfolding process showed a spectral shape similar to the input spectrum, but scaled by a factor of 0.32. This scaling factor is consistent with the calculated to measured ratios observed. Using the flux-to-dose-rate conversion function, the dose rate metric between the scaled unfolded spectrum and the default spectrum is 6%. The calculated-to-measured ratios after unfolding (using the unfolded spectrum) were closer to unity, with 2 of 5 error bars overlapping a value of 1. These results were published in the AccApp’11 conference proceedings [75]. Because only 2 of 5 error bars overlapped the value of 1, because of the spread in the calculated-to-measured ratios, and because the output spectrum appeared suspiciously close to the input spectrum, a complete re-analysis of the data was done. This re-analysis uncovered errors, and therefore the unfolding produced a different result.

90 (a) Unfolded Reaction Rate to Measured Reaction Rate Ratio

(b) Calculated and Unfolded Source Spectra

Figure 5.19: The unfolded reaction rate to measured reaction rate ratio after initial unfolding and the initial unfolded spectrum as compared to the calculated spectrum. The calculated spectrum is scaled by an overall factor of 0.32 for the best ﬁt to the measured data.

91 Reanalyzed

The reanalyzed ARCS spectrum used reaction rates from 6 measured nuclide activities: 206Bi, 205Bi, 204Bi, 203Bi, 58Co, 24Na. Because the time analysis results of the candidate peaks were inconsistent with 202Bi, this nuclide was eliminated from the reanalysis. The time analysis results showed the candidate peaks for 24Na were good candidates, so 24Na was added to the reanalysis. Due to the relatively long half-life of 58Co, the time analysis results were inconclusive, but 58Co was included in the reanalysis. Errors in the self-shielding factors were discovered for the bismuth nuclides in the initial analysis. These errors were due to poor interpolation of values calculated for the self-shielding factors. In the reanalysis, the self-shielding factors were computed for the exact values of the gamma-ray energies. MAXED was again used for the unfolding process. The computed ARCS spectrum was used for the initial default spectrum and for determining the response functions. The output spectral shape diﬀered noticeably from the input spectral shape, and was scaled by 0.32. This factor of 0.32 indicates that the calculated spectrum was more intense than the unfolded spectrum. The calculated-to-measured ratios after unfolding were all consistent with unity, and all error bars overlapped 1. This result indicates that this unfolded spectrum is more consistent with the measurements, and the re-analysis is more reliable than the initial unfolding.

92 (a) Unfolded Reaction Rate to Measured Reaction Rate Ratio

(b) Calculated and Unfolded Source Spectra

Figure 5.20: The unfolded reaction rate to measured reaction rate ratio after reanalyzed unfolding and the unfolded spectrum as compared to the calculated spectrum. The calculated spectrum is scaled by an overall factor of 0.32 for the best ﬁt to the measured data.

93 Figure 5.21: Beam power during the June 22, 2011 POWGEN foil irradiation.

5.2 June 22, 2011 POWGEN

5.2.1 Foil Irradiation

Foil Packet

The foil packet used at the POWGEN beamline consisted of nine types of foils: 3 nickel foils (previously irradiated), and a single bismuth foil in a top stack; 1 copper, 1 aluminum, 1 gold, 1 iron, 1 indium, 1 titanium, and 1 cobalt foil in a bottom stack. Five new foil types (as compared to the ARCS foil packet) were used because the additional foils provided reaction cross sections that would overlap and span the range of energies. The foils were mounted to an aluminum holder and wrapped in aluminum foil. The packet was mounted to a sample holder and placed in the beam at the sample location of the POWGEN instrument.

94 Beam Power

The beam power history during the POWGEN irradiation was fully archived, but fluctuated significantly. Initially the beam power level was approximately 810 kW. While at this level, the beam was off for several intervals of a few minutes. At approximately, 05:30 the beam power underwent a gradual rise, before suddenly dropping to half-power. Over the next several minutes, the power rose gradually, and then dropped to zero. After a several minute time interval of no beam, beam power returned to less than half the original power level. For a period of approximately two hours, the power level fluctuated variously between 200 kW and 400 kW. At 07:45, the beam was turned off. The foils were removed from the sample position shortly after 09:00. The average power from the start of the irradiation at 22:06 to the time beam turned off at 07:45 is 682.794 kW. Due to the power fluctuations, the power history corrections are significant.

Irradiation Time

The irradiation began at 22:06 and ended when the beam shut oﬀ at 07:45. The total irradiation time was 34740 ± 120 s (∼ 9.65 hours).

Cooling Time

The cooling time started at 07:45 and ended at 09:55 for a total cooling time of 7800 s ± 120 s (∼ 2 hours 10 minutes).

5.2.2 Autoradiography

After irradiation and counting, the foils (except the bismuth foil) were placed on a gamma- ray sensitive image plate for autoradiography. The foils and plate needed to be sealed in a light-tight case, which was too small for the 1 cm thick bismuth foil, so it was excluded. The foils were allowed to activate the image plate undisturbed for ∼1 week, and then the plate was read using the Fuji Medical Systems FLA-7000. The autoradiograph showed partial irradiation of the gold and indium foils, and low intensity total irradiation of the nickel foils. The nickel foils, having been previously irradiated, retained a residual amount

95 (a)

(b)

Figure 5.22: Autoradiograph for the foils irradiated at the POWGEN beamline. of 60Co activity. This residual 60Co activity from the previous irradiation is likely the source of the image on the autoradiograph. Although all foils in the top row of the radiograph image were in the same stack in the foil packet, only the gold and indium foils were visible. The activity of these foils during the exposure period must have been too low to be measured by the image plate.

96 Figure 5.23: Energy calibration curve for the HPGe spectrometer used for the POWGEN foil irradiation.

5.2.3 Gamma Spectrometry

Energy and Eﬃciency Calibrations

Energy and efficiency calibrations were performed using the following radioisotope calibration sources: 22Na, 109Cd, 57Co, 113Sn, 137Cs, 88Y. The energy calibration used 9 peaks ranging from 88 keV to 1836 keV. A linear fit was made to the data with a coefficient of determination r2 value of 0.999. The efficiency calibration used 14 peaks ranging from 160 keV to 1835 keV. A linear fit was made to the data (natural log of efficiency vs. natural log of energy) with a coefficient of determination r2 value of 0.999. Unlike the ARCS measurement, the calibrations and samples were measured using the sample holder sufficiently far from the detector end cap (∼10 cm) to minimize TCS. Efficiency calibrations below 160 keV were performed, but these data were not needed to analyze the gamma-ray spectrum.

97 Figure 5.24: Eﬃciency calibration curve for the HPGe spectrometer used for the POWGEN foil irradiation.

Count Time

The count used for the analysis was started at 09:55:48. Total Real Time: 541,748.00 ± 1 s (∼ 150.5 hours) Total Live Time: 528,300.00 ± 1 s (∼ 146.8 hours)

98 Measured Candidate Peak and Expected Peaks Not Measured

The results of the gamma-ray spectrometry showed several peaks, but none of the peaks are due to threshold reactions. Several of the peaks were identified as background peaks, and the remainder were due to (n,γ) reactions on the indium and gold foils, and the residual 60Co activity of the nickel foils (Table 5.3). This measurement is consistent with the autoradiograph, which showed no or low activity on all foils except gold, indium, and nickel. The gamma-ray spectra data are inconsistent with a high-energy flux irradiation of the foils. Using the calculated POWGEN flux spectrum, many threshold reaction peaks were expected. The expected peaks not measured and their expected counts are listed in Table 5.4. The possible source of errors are discussed in the Section 6.2.

Nuclide Peak Energy Nuclide Peak Energy (Foil) keV (Background) keV 116mIn 417.00 Annihilation 511.39 116mIn 1097.36 116mIn 1293.69 40K 1460.54 116mIn 1507.69 116mIn 2112.24 214Bi 1764.41 214Bi 1847.1 198Au 411.69 214Bi 2203.05 198Au 675.89 198Au 1087.98 208Tl 2615.72

60Co 1173.54 60Co 1332.57

Table 5.3: Candidate peaks

99 Nuclide Peak Energy Expected Nuclide Peak Energy Expected keV Counts keV Counts 195Au 98.85 73 56Mn 846.77 1441 115mIn 336.24 30700 204Bi 899.15 5768 196Au 355.68 5973 202Bi 960.67 597 201Bi 629.1 212 48Sc 983.52 340 206Bi 803.1 21049 46Sc 1120.55 313 58Co 810.76 454 24Na 1368.63 973 203Bi 820.3 1761 205Bi 1764.36 806 54Mn 834.85 451

Table 5.4: Expected peaks

Figure 5.25: The POWGEN gamma spectrum

100 Figure 5.26: The POWGEN neutron source spectrum

5.2.4 Unfolding

Because threshold reaction activities were not measured, spectral unfolding could not be conducted for these data. The calculated POWGEN source spectrum that would have been used as the default spectrum and to generate the response functions is shown in Figure 5.26.

101 Chapter 6

Conclusions

The chapter presents conclusions drawn from the measurement results. Future work for improved measurements are also discused.

6.1 ARCS Measurements Conclusions

Based on the reanalyzed data and unfolding, the measured high-energy source term spectrum at ARCS is about 0.32 the calculated source term, therefore the measured flux is approximately 1/3 the calculated flux. The measured spectral shape differs noticeably from the calculated spectrum in the range of 25 MeV to 80 MeV. This energy range is where the computational models used to calculate the spectrum are known to perform relatively poorly [76]. Based on the calculated-to-measured ratios, the unfolded spectrum predicts the six reaction product activities precisely. Therefore the reanalyzed unfolded ARCS spectrum is should be regarded as an internally consistent measurement, which has yet to be validated by further measurements. It is a reasonable first step in the full measurement of the SNS high-energy flux spectrum, but it is not incontrovertible. Future and repeated measurements are necessary to draw conclusions with high confidence, but these measurements and analysis do not show gross disagreement with the calculated high-energy flux spectrum. These measurements indicated that a conservative source term was used for the shielding design. The measurements performed at ARCS informed several improvements for sub-

102 sequent measurements. The gamma-ray spectrometry count time was limited by the use of a 7 L LN2 dewar. This dewar could not be refilled without removing it from the shield, thus causing a change in geometry and requiring recalibration. A larger 30 L dewar that could be refilled on-line replaced the smaller Dewar. This allowed the count time for the POWGEN foils to be longer by a factor of > 3.5. Experience with TCS issues and source calibration geometry also informed the subsequent measurements. At first, it was assumed that close measurement geometry would provide the best data because of the high geo- metric efficiency and high count rates. Only after several poor calibration curves were made was it realized that close geometry is difficult for accurate efficiency calibrations. For the POWGEN foil measurements, a sample holder was designed and used. This virtually eliminated TCS issues and enforced standardization of the calibration geometry with the sample geometry. The ARCS measurements also informed the use of additional foils in the POW- GEN measurements. Foils were added which offered threshold reactions spanning a broader energy range (e.g. 115In(n,n’)115mIn with a threshold at 340 keV) and reactions that overlapped other reactions (e.g gold (n,xn) reactions which overlap bismuth (n,xn) reactions). The ARCS gamma-ray spectrum file was saved several times during the count, which later proved to be useful for doing the time analysis and decay constant measurement. For the POWGEN measurement, periodic saving of the gamma-ray spectrum occurred every 15 minutes by a software script to improve the time analysis.

6.2 POWGEN Measurements Conclusions

The POWGEN foils gamma-ray spectrum results and the autoradiograph results point to an error in the experimental setup. Therefore, nothing can be concluded about the high- energy neutron ﬂux spectrum from these data. It is useful for planning future measurements to consider what experimental error(s) are consistent with the observed data. The source of the experimental error may be deduced and avoided in future measurements The gamma-ray spectrum and the autoradiograph indicated activation of the gold and indium foils. The gold and indium foils were in a stack that also included aluminum, cobalt, copper, iron, and titanium foils, but no activity from these foils was

103 measured with the gamma-ray spectrometer or autoradiograph. The gold and indium foils showed partial irradiation (a non-uniform spatial distribution) on the autoradiograph, and from the gamma-ray spectrum the origin of their activity is due to (n,γ) reactions, not threshold reactions. One conceivable scenario is that the foils were exposed to a beam of thermal energy neutrons, but not a beam of high-energy neutrons. This is possible because a beam guide is in use at POWGEN. The beam guide directs thermal energy neutrons, but has no impact on high-energy neutrons, therefore two beam “footprints” are possible at the sample location. The foils could have been inside the thermal footprint, but not inside the high-energy footprint, thus thermal, but not high-energy, activation would have occurred. But this explanation is contradicted by other data. The lithium phosphate beam slits were reportedly fully closed, so no thermal beam should have been present. The position of the beam slits was confirmed in the data archive, although it is conceivable that the slit positions are false indications. If the slits were actually open and thermal energy beam did irradiate the foil stack, 60Co from the cobalt foil (by the 59Co(n,γ)60Co reaction) should have been detected in the gamma-ray spectrum and on the autoradiograph. While 60Co was detected in the gamma-ray spectrum, this is due entirely to the residual activity of the nickel foil. When measured separately, the cobalt foil showed no activity. Based on the evidence, thermal beam activation is not a consistent explanation. Another conceivable explanation is that there is no measurable high-energy flux at the POWGEN sample location. If all sources of experimental error were eliminated, this explanation would be the most straightforward. And while this explanation cannot be negated, based on the expected counts of Table 5.4, the calculated POWGEN source term would be incorrect by orders of magnitude. For example, the highest expected number of counts was 30,000 due to 115mIn. All other factors remaining unchanged, if the spectrum was scaled by 0.001, a measurable number of counts should have been present in the gamma-ray spectrum. Similar arguments could be made for the expected counts due to 206Bi and 196Au. So while it is possible that high-energy flux is not present at POWGEN, the calculated spectrum would need to be grossly erroneous. “Room return” is the phenomenon whereby radiation, typically neutrons, are reflected back from surrounding material. This effectively makes the walls, floors, and

104 ceiling of the beam room an additional source to consider. It is possible, that the foils were not exposed to any beam, and they were instead activated by the room return neutrons. Without modeling the surrounding structure, the magnitude of the room return cannot be easily gauged. While room return could explain the absence of high energy neutron activation on the foils, it cannot explain why only the gold and indium foils were activated. The room return explanation is also contradicted by the partial irradiation indicated on the autoradiograph. Room return is typically ambient, so a uniform irradiation of the foils is expected. The most convincing explanation of the spectrometry and autoradiograph results is a failure with the T0 chopper. The T0 chopper, a large rotating block of inconel, is designed to stop the high-energy neutrons from traveling to the sample location. The

T0 chopper was allegedly stopped in the open position, allowing high-energy neutrons to pass. If the T0 chopper continued to operate during these measurements, it would have blocked the high-energy neutrons, and the lithium phosphate slits would have blocked the thermal-energy neutrons. Only a beam of resonance-energy neutrons would have arrived at the sample site to irradiate the foils. These resonance-energy neutrons would have caused activation in the gold and indium (due to their relatively high resonance integrals), but not aﬀected the other foils. The resonance-energy neutrons would be too high in energy for signiﬁcant activation of cobalt through the 59Co(n,γ)60Co reaction (which has a low resonance integral), and they would be too low for threshold activation. While qualitatively

this T0 chopper failure theory appears to provide the most complete explanation, further modeling is required to provide a quantitative assessment of the theory.

6.3 Future Work

These measurements are the ﬁrst in the attempt to quantify the high-energy neutron ﬂux at the SNS. They have provided a basis to conduct future measurements, but several improvements can be made. From an administrative perspective, the immediate objective of future work should be to preserve the knowledge advanced by this work. The entire method from foil selection, to irradiation, to gamma-ray spectrometric and autoradiography, to computa-

105 Figure 6.1: Exploded side view of a foil packet designed to address issues encountered with the POWGEN foil irradiation. tional analysis needs to be codified into a repeatable process. While this document serves as a detailed guide to the work completed, more concise operation documentation is required. Several “HowTo” documents have been written, but these documents need to be vetted and revised. The computational work was conducted mostly on a personal computer, so software and auxiliary programs need to be transferred to systems accessible by SNS staff. From an experimental standpoint, the next measurement should take place with a foil packet already designed and constructed for use. This foil packet attempts to address some of the issues encountered with the POWGEN foil irradiation (Figure 6.1). Two aluminum brackets form the structure of the packet, with metal fiducials (nuts on screws) mounted to the front. The metal fiducials are designed to cast bam shadows on the large (∼ 12 × 12 cm) aluminum foils behind them, but not on the high-energy foil packet. The aluminum foils have enough area to fully contain the beam footprint, and they will be used for autoradiography immediately following irradiation. A cadmium plate (0.6 mm thick) is used as a thermal neutron cover to shield the high-energy foil packet and the rear

106 aluminum foil. The high-energy foil packet contains 9 types of foils and is similar to the packet used for the POWGEN measurement. Autoradiography on the rear aluminum foil should produce an image of the high-energy beam footprint (since it’s behind the cadmium cover), while autoradiography on the forward aluminum foil should produce a footprint of both high-energy and thermal beam. The entire foil packet can be optionally used with a neutron-sensitive image plate placed to the rear for direct radiography of the beam footprint. The newly constructed foil packet eliminates concerns of false indications of the lithium phosphate beam slits because the cadmium cover provides a local thermal beam filter. The large aluminum foils allow for radiography to commence in parallel with the gamma-ray spectrometry. The two foils would show any difference in location of the thermal and high-energy beam footprints. If this packet had been used for the POWGEN foil irradiation, it is likely that several additional data would have been used to deduce the source of the experimental error. A second measurement at the ARCS beamline should be a priority. With more foils, higher beam power, longer irradiation, and more experience a second measurement would provide validation (or invalidation) of the unfolded spectrum already measured. If possible, two measurements should occur within a short time interval to minimize the likelihood of changes in conditions. From an analysis perspective, the priority should be to unfold the spectrum using alternative algorithms. The most obvious choice is GRAVEL, since the software is already available with UMG. The IQU program could be used to propagate uncertainties from the activity measurements to the unfolded spectra. Additional metrics could be designed to judge the conformity of an output spectrum with a default spectrum. One example metric could be the difference in activity the two spectra produce in a sample material (e.g. stainless steel, concrete). Beyond these near-term objectives, several broad goals can be envisioned. Mea- surement of the high-energy neutron flux using alternative means, such as a proton recoil telescope, could be used to confirm measurements made with foil activation. A software suite that merges response function generation, data collection of gamma-ray spectrometry, calculational corrections, and spectral unfolding could be produced.

107 6.4 General Conclusions

It was the goal of this work to develop a foil activation method to measure high-energy neutron flux spectra at the SNS by researching the scientific literature, assembling an experimental apparatus, performing experiments, analyzing the results, and refining the technique based on experience. This goal has been achieved. A method has been constructed, measurements were conducted, and techniques refined. The results remain to be validated by additional measurements, but this work has provided a framework and a foundation to conduct those future measurements. Personally speaking, conducting this work has been enriching. When I began, I had classroom knowledge of topics like gamma-ray spectrometry and transport theory. My skills using MCNPX were basic, my knowledge of nuclear data was minimal, and my experimental experience was limited. I have traveled down many mental blind alleys while working on this topic. I have spent futile hours trying to track down the source of so many errors - computational, experimental, calculational. At times, it seemed impossible. But, also I’ve had small eureka moments when the code ran, the numbers agreed, and the measurement worked. Over time, I noticed that the number of mistakes I made has declined, and when I do make errors, I know how to recover from them faster. My skills and knowledge in spectrometry, transport theory, MCNPX, nuclear data, etc. have improved, but the most significant lesson has come from going through the entire process. This work accomplished a small engineering task at the Spallation Neutron Source, but it also provided me with an invaluable education and a deeply rewarding experience.

108 Bibliography

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117 Appendix A

Units and Constants

Units (Conversions to Syst´emeInternational base units are provided.) b barn = 10−28 m2

C Coloumb = 1 Ampere · s

10 decays Ci Curie = 3.7 × 10 s

−19 kg·m2 eV electron-volt = 1.602176565(3) × 10 s g gram = 10−3 kg

cycles Hz Hertz = 1 s in inches = 2.54 × 10−2 m

K Kelvin

L Liter = 10−3 m3 m meter s second

kg·m2 W Watt = 1 s3

Constants

−23 kg·m2 k = 1.3806488(13) × 10 s·K (Boltzmann Constant)

118 Appendix B

Experimental Foils

119 Image Material Beamlines Radius Mass Thickness (measured) (measured) (calculated) in cmg cm

0.175(2) ARCS 0.174(5) Aluminum POWGEN 0.64(0) 0.175(7) 0.0498

ARCS 7.581(8) Bismuth POWGEN 0.49(5) 7.581(9) 1.00

Cobalt POWGEN 0.5(0) 0.173(9) 0.0249

ARCS Copper POWGEN 0.64(0) 0.152(1) 0.0132

Gold POWGEN 0.5(0) 0.369(1) 0.0243

Indium POWGEN 0.5(0) 0.570(1) 0.0993

Iron POWGEN 0.5(0) 0.613(6) 0.0991

0.014(5) ARCS 0.014(0) Nickel POWGEN 0.64(0) 0.013(8) 0.0012

Titanium POWGEN 0.5(0) 0.350(4) 0.0989

Table B.1: Foils irradiated at ARCS and POWGEN. Where multiple mass values are listed, each value indicates a single foil.

120 Appendix C

Foil Gamma-ray Self-Shielding Factors

121 Figure C.1: Gamma-ray self-shielding factors for the aluminum foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.9600 0.007 950 0.9830 0.004 150 0.9650 0.006 1000 0.9830 0.004 200 0.9680 0.006 1100 0.9840 0.004 250 0.9690 0.006 1200 0.9850 0.004 300 0.9710 0.006 1300 0.9850 0.004 350 0.9720 0.005 1400 0.9850 0.004 400 0.9740 0.005 1500 0.9860 0.004 450 0.9770 0.005 1600 0.9860 0.004 500 0.9770 0.005 1700 0.9860 0.004 550 0.9790 0.005 1800 0.9860 0.004 600 0.9790 0.005 1900 0.9870 0.004 650 0.9810 0.004 2000 0.9870 0.004 700 0.9810 0.004 2100 0.9870 0.004 750 0.9810 0.004 2200 0.9870 0.004 800 0.9810 0.004 2300 0.9870 0.004 850 0.9810 0.004 2400 0.9870 0.004 900 0.9820 0.004 2500 0.9880 0.004

Table C.1: Gamma-ray self-shielding factors for the aluminum foil listed in Table B.1. Error bars represent absolute standard error.

122 Figure C.2: Gamma-ray self-shielding factors for the bismuth foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.0281 0.016 950 0.7460 0.013 150 0.0770 0.013 1000 0.7535 0.013 200 0.1491 0.011 1100 0.7710 0.012 250 0.2367 0.011 1200 0.7815 0.012 300 0.3292 0.011 1300 0.7875 0.012 350 0.4113 0.011 1400 0.7920 0.016 400 0.4749 0.011 1500 0.8050 0.016 450 0.5341 0.011 1600 0.8090 0.015 500 0.5813 0.011 1700 0.8100 0.015 550 0.6142 0.011 1800 0.8180 0.015 600 0.6493 0.012 1900 0.8200 0.015 650 0.6720 0.011 2000 0.8230 0.015 700 0.6933 0.012 2100 0.8250 0.015 750 0.7063 0.012 2200 0.8250 0.015 800 0.7220 0.011 2300 0.8260 0.015 850 0.7280 0.014 2400 0.8280 0.014 900 0.7340 0.014 2500 0.8270 0.015

Table C.2: Gamma-ray self-shielding factors for the bismuth foil listed in Table B.1. Error bars represent absolute standard error.

123 Figure C.3: Gamma-ray self-shielding factors for the cobalt foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.8620 0.013 950 0.9680 0.006 150 0.9280 0.009 1000 0.9680 0.006 200 0.9440 0.008 1100 0.9690 0.006 250 0.9500 0.007 1200 0.9730 0.005 300 0.9560 0.007 1300 0.9740 0.005 350 0.9580 0.007 1400 0.9750 0.005 400 0.9600 0.007 1500 0.9770 0.005 450 0.9600 0.007 1600 0.9780 0.005 500 0.9620 0.006 1700 0.9780 0.005 550 0.9620 0.006 1800 0.9780 0.005 600 0.9650 0.006 1900 0.9780 0.005 650 0.9650 0.006 2000 0.9780 0.005 700 0.9650 0.006 2100 0.9780 0.005 750 0.9660 0.006 2200 0.9780 0.005 800 0.9680 0.006 2300 0.9780 0.005 850 0.9680 0.006 2400 0.9820 0.004 900 0.9680 0.006 2500 0.9820 0.004

Table C.3: Gamma-ray self-shielding factors for the cobalt foil listed in Table B.1. Error bars represent absolute standard error.

124 Figure C.4: Gamma-ray self-shielding factors for the copper foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.9080 0.010 950 0.9810 0.004 150 0.9480 0.007 1000 0.9810 0.004 200 0.9620 0.006 1100 0.9820 0.004 250 0.9660 0.006 1200 0.9830 0.004 300 0.9680 0.006 1300 0.9830 0.004 350 0.9720 0.005 1400 0.9840 0.004 400 0.9730 0.005 1500 0.9840 0.004 450 0.9730 0.005 1600 0.9840 0.004 500 0.9730 0.005 1700 0.9840 0.004 550 0.9740 0.005 1800 0.9850 0.004 600 0.9740 0.005 1900 0.9850 0.004 650 0.9760 0.005 2000 0.9850 0.004 700 0.9790 0.005 2100 0.9850 0.004 750 0.9790 0.005 2200 0.9860 0.004 800 0.9800 0.005 2300 0.9860 0.004 850 0.9800 0.005 2400 0.9860 0.004 900 0.9810 0.004 2500 0.9880 0.004

Table C.4: Gamma-ray self-shielding factors for the copper foil listed in Table B.1. Error bars represent absolute standard error.

125 Figure C.5: Gamma-ray self-shielding factors for the gold foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.2169 0.016 950 0.9380 0.008 150 0.4553 0.013 1000 0.9400 0.008 200 0.6240 0.012 1100 0.9440 0.008 250 0.7210 0.011 1200 0.9460 0.008 300 0.7705 0.012 1300 0.9500 0.007 350 0.8100 0.011 1400 0.9530 0.007 400 0.8400 0.014 1500 0.9540 0.007 450 0.8630 0.013 1600 0.9560 0.007 500 0.8810 0.012 1700 0.9550 0.007 550 0.8950 0.011 1800 0.9570 0.007 600 0.9080 0.010 1900 0.9570 0.007 650 0.9160 0.010 2000 0.9580 0.007 700 0.9210 0.009 2100 0.9590 0.007 750 0.9210 0.009 2200 0.9590 0.007 800 0.9260 0.009 2300 0.9590 0.007 850 0.9340 0.008 2400 0.9590 0.007 900 0.9360 0.008 2500 0.9580 0.007

Table C.5: Gamma-ray self-shielding factors for the gold foil listed in Table B.1. Error bars represent absolute standard error.

126 Figure C.6: Gamma-ray self-shielding factors for the indium foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.4147 0.010 950 0.9390 0.008 150 0.6608 0.010 1000 0.9390 0.008 200 0.7745 0.012 1100 0.9410 0.008 250 0.8235 0.010 1200 0.9460 0.008 300 0.8540 0.013 1300 0.9470 0.008 350 0.8670 0.012 1400 0.9470 0.008 400 0.8840 0.012 1500 0.9470 0.008 450 0.8960 0.011 1600 0.9480 0.007 500 0.9100 0.010 1700 0.9500 0.007 550 0.9200 0.009 1800 0.9540 0.007 600 0.9220 0.009 1900 0.9540 0.007 650 0.9240 0.009 2000 0.9540 0.007 700 0.9280 0.009 2100 0.9530 0.007 750 0.9300 0.009 2200 0.9530 0.007 800 0.9350 0.008 2300 0.9530 0.007 850 0.9380 0.008 2400 0.9540 0.007 900 0.9360 0.008 2500 0.9570 0.007

Table C.6: Gamma-ray self-shielding factors for the indium foil listed in Table B.1. Error bars represent absolute standard error.

127 Figure C.7: Gamma-ray self-shielding factors for the iron foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.7463 0.011 950 0.9320 0.009 150 0.8220 0.015 1000 0.9340 0.008 200 0.8570 0.013 1100 0.9350 0.008 250 0.8670 0.012 1200 0.9380 0.008 300 0.8860 0.011 1300 0.9400 0.008 350 0.8940 0.011 1400 0.9420 0.008 400 0.9000 0.011 1500 0.9450 0.008 450 0.9060 0.010 1600 0.9470 0.008 500 0.9090 0.010 1700 0.9470 0.008 550 0.9150 0.010 1800 0.9470 0.008 600 0.9180 0.010 1900 0.9480 0.007 650 0.9210 0.009 2000 0.9480 0.007 700 0.9240 0.009 2100 0.9480 0.007 750 0.9250 0.009 2200 0.9500 0.007 800 0.9270 0.009 2300 0.9530 0.007 850 0.9270 0.009 2400 0.9530 0.007 900 0.9320 0.009 2500 0.9530 0.007

Table C.7: Gamma-ray self-shielding factors for the iron foil listed in Table B.1. Error bars represent absolute standard error.

128 Figure C.8: Gamma-ray self-shielding factors for the nickel foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.9870 0.004 950 0.9980 0.001 150 0.9930 0.003 1000 0.9980 0.001 200 0.9950 0.002 1100 0.9980 0.001 250 0.9950 0.002 1200 0.9980 0.001 300 0.9970 0.002 1300 0.9980 0.001 350 0.9970 0.002 1400 0.9980 0.001 400 0.9970 0.002 1500 0.9980 0.001 450 0.9970 0.002 1600 0.9990 0.001 500 0.9970 0.002 1700 1.0000 0.000 550 0.9970 0.002 1800 1.0000 0.000 600 0.9970 0.002 1900 1.0000 0.000 650 0.9970 0.002 2000 1.0000 0.000 700 0.9970 0.002 2100 1.0000 0.000 750 0.9970 0.002 2200 1.0000 0.000 800 0.9970 0.002 2300 1.0000 0.000 850 0.9980 0.001 2400 1.0000 0.000 900 0.9980 0.001 2500 1.0000 0.000

Table C.8: Gamma-ray self-shielding factors for the nickel foil listed in Table B.1. Error bars represent absolute standard error.

129 Figure C.9: Gamma-ray self-shielding factors for the titanium foil listed in Table B.1. Error bars represent absolute standard error.

Gamma-ray Self-Shielding MCNPX Gamma-ray Self-Shielding MCNPX Energy Factor RSE Energy Factor RSE keV keV 100 0.8470 0.013 950 0.9590 0.007 150 0.9000 0.011 1000 0.9600 0.007 200 0.9230 0.009 1100 0.9630 0.006 250 0.9330 0.009 1200 0.9640 0.006 300 0.9380 0.008 1300 0.9640 0.006 350 0.9410 0.008 1400 0.9660 0.006 400 0.9430 0.008 1500 0.9670 0.006 450 0.9450 0.008 1600 0.9690 0.006 500 0.9470 0.008 1700 0.9690 0.006 550 0.9470 0.008 1800 0.9690 0.006 600 0.9480 0.007 1900 0.9720 0.005 650 0.9470 0.008 2000 0.9730 0.005 700 0.9520 0.007 2100 0.9730 0.005 750 0.9520 0.007 2200 0.9730 0.005 800 0.9520 0.007 2300 0.9730 0.005 850 0.9530 0.007 2400 0.9740 0.005 900 0.9570 0.007 2500 0.9760 0.005

Table C.9: Gamma-ray self-shielding factors for the titanium foil listed in Table B.1. Error bars represent absolute standard error.

130 Appendix D

Reaction Cross Section Plots

131 Figure D.1: Threshold reaction cross sections from the TENDL-2010 library for aluminum.

132 Figure D.2: Threshold reaction cross sections from the TENDL-2010 library for bismuth.

133 Figure D.3: Threshold reaction cross sections from the TENDL-2010 library for cobalt.

134 Figure D.4: Threshold reaction cross sections from the TENDL-2010 library for copper.

135 Figure D.5: Threshold reaction cross sections from the TENDL-2010 library for gold.

136 Figure D.6: Threshold reaction cross sections from the TENDL-2010 library for indium.

137 Figure D.7: Threshold reaction cross sections from the TENDL-2010 library for iron.

138 Figure D.8: Threshold reaction cross sections from the TENDL-2010 library for nickel.

139 Figure D.9: Threshold reaction cross sections from the TENDL-2010 library for titanium.

140 Vita Nicholas Luciano is from Englewood, Florida. His interest in nuclear engineering began when he was a student assistant at the Neely Nuclear Research Center at the Georgia Institute of Technology. After he received his B.S. in Physics from Georgia Tech, he worked as a physics associate at the Collider-Accelerator Department at Brookhaven National Lab- oratory. In Brookhaven’s main control room, Nick operated several particle accelerators, including the Relativistic Heavy Ion Collider and the Alternating Gradient Synchrotron, two of the world’s largest hadron accelerators. It was in the main control room that he met his future fiancée,Jennifer Niedziela. He was also part of a team that built and tested a prototype hydrogen jet luminescence monitor for 100 GeV ion beam profile measurements. In 2006, he joined the Spallation Neutron Source staff as a control room shift supervisor. While working at SNS, he assisted the ion source group in developing a new hydrogen ion extraction and transport system. Nick began taking part-time classes in nuclear engineering at the University of Tennessee, and soon after, Jennifer moved to SNS as a graduate student in physics at UT. In the fall of 2010, both Nick and Jen left their jobs at SNS to become full-time graduate students. They are engaged to be married in 2012. Nick is currently pursuing his Ph.D. in nuclear engineering at UT, and hopes to continue research after completing his degree.

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